Maths - Basic Analysis Result

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BrandonS15
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#1
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#1
Good Morning,

I’m currently reading through the STEP preparation book and I’ve come across this idea which I haven’t actually seen before, its a basic analysis statement stating this:
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So to better understand it, I’ve tried to use an example and see if it works and currently I’m not getting a correct result, using an example I yield k/2 as the maximum area whereas the analysis statement gives k/4 so where have I supposedly gone wrong here?
Workings:
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Many thanks in advance.
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ghostwalker
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#2
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(Original post by BrandonS15)
Good Morning,

I’m currently reading through the STEP preparation book and I’ve come across this idea which I haven’t actually seen before, its a basic analysis statement stating this:
Name:  2B295B9C-319B-47F7-BDAB-F16AE703AE86.jpg.jpeg
Views: 7
Size:  15.5 KB
So to better understand it, I’ve tried to use an example and see if it works and currently I’m not getting a correct result, using an example I yield k/2 as the maximum area whereas the analysis statement gives k/4 so where have I supposedly gone wrong here?
Workings:
Name:  D33D2BF0-2693-42F3-B08C-BB0FAEFD845C.jpg.jpeg
Views: 5
Size:  24.8 KB

Many thanks in advance.

Where did your b-a=k/4 come from - on the extremem right?
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BrandonS15
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#3
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(Original post by ghostwalker)
Where did your b-a=k/4 come from - on the extremem right?
I’m equating the given area from the statement (b-a)k and the computed area (1/4k)k hence getting b-a =1/4k if the statement holds then they would equal, so I’m guessing I’ve made a mistake somewhere?
b is the upper bound while a is the lower bound as suggested in the statement for analysis
Last edited by BrandonS15; 1 year ago
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ghostwalker
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(Original post by BrandonS15)
I’m equating the given area from the statement (b-a)k and the computed area (1/4k)k hence getting b-a =1/4k if the statement holds then they would equal, so I’m guessing I’ve made a mistake somewhere?
b is the upper bound while a is the lower bound as suggested in the statement for analysis
But they're not equal.

Your actual area under the curve between 0 and (k-6)/2 is \frac{k^2}{4}

The analysis statement says the area is less than (b-a)k which works out to \frac{k^2}{2}


This diagram may help:

The analysis statement gives the area in the green rectangle. And the integral gives the area shaded in red.

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Last edited by ghostwalker; 1 year ago
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BrandonS15
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#5
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(Original post by ghostwalker)
But they're not equal.

Your actual area under the curve between 0 and (k-6)/2 is \frac{k^2}{4}

The analysis statement says the area is less than (b-a)k which works out to \frac{k^2}{2}


This diagram may help:

The analysis statement gives the area in the green rectangle. And the integral gives the area shaded in red.

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I see, doesn’t the analysis statement give the minimum and maximum possible area for f(x) under its function when the range is restricted from 0 to k? So using my example of f(x) I don’t understand how what I’ve computed isn’t the maximum area which should equal the maximum area given by the analysis statement?
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ghostwalker
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(Original post by BrandonS15)
I see, doesn’t the analysis statement give the minimum and maximum possible area for f(x) under its function when the range is restricted from 0 to k? So using my example of f(x) I don’t understand how what I’ve computed isn’t the maximum area which should equal the maximum area given by the analysis statement?
I edited my previous post to include a diagram, which might help clarify things.
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BrandonS15
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#7
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(Original post by ghostwalker)
I edited my previous post to include a diagram, which might help clarify things.
Doesn’t the integral in the analysis statement mean area under f(x) between a and b and so wouldn’t it also be the red area? As thats what my current understanding it and therefore I’m letting the two equal in my workings, if not then I’m not understanding why the analysis statement doesn’t mean this because its saying the area between bounds a and b can go from a minimum of 0 to a maximum of (b-a)k and using my example I have computed the maximum area under f(x) as per the analysis statement which leads me to the contradiction as the end
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ghostwalker
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#8
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(Original post by BrandonS15)
Doesn’t the integral in the analysis statement mean area under f(x) between a and b and so wouldn’t it also be the red area? As thats what my current understanding it and therefore I’m letting the two equal in my workings, if not then I’m not understanding why the analysis statement doesn’t mean this because its saying the area between bounds a and b can go from a minimum of 0 to a maximum of (b-a)k and using my example I have computed the maximum area under f(x) as per the analysis statement which leads me to the contradiction as the end
Yes, the red area represents the area under f(x) between a and b, in the analysis statement.

BUT, the analysis statement is saying this area is less than or equal to the area of the rectangle; the width (a times b) times the height (k).

You've chosen an inteval that depends on k. That's not a requirement of the analysis statement.

The statement itself, is intuitively obvious. It's saying, if k is greater than the maximum height of the curve, then the area under the curve is less than the width of the inverval times k.

For the area to go from 0 to the maximum of (b-a)k you'd need to chose different functions.

For a given a,b,k, to get the minimum value you'd need to choose y=0, and to get the maximum value you'd choose y=k as your functions.
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