Maths Surge and Logistics PDF

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Surge and Logistics...

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MATHEMATICAL METHODS

SURGE AND LOGISTIC FUNCTIONS

SACE ID: 893239E RASOUL HEIDARI

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SACE ID: 893239E Surge and Logistic Models Introduction: The aim of this report is to investigate surge and logistic models and how they incorporate in real life applications. Surge models are models in the form of −bx f ( x ) = Ax e ,t ≥ 0 and are used in real life applications such as the alcohol level in a human versus the hours since consumption. Surge models present this through the rapid increase in concentration and the slow decrease in concentration. Logistic models are in the form

P (t )=

L 1+ Ae−bt

and model growth such as the

total number of people that have the flu. Logistic models present this through the slow increase at the beginning, shifting to a rapid growth and then slowing down before levelling off.

Part 1: The Surge Function A Surge function is in the form constants. Graph

y=f (x)

and

−bx

f ( x )= Axe

where A and b are positive

y=f '(x ) where A = 10 and b = 4

Therefore, the c To determine the stationary points, we let f’(x) = 0

To determine the Inflection points for inflection point = when

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y coordinate of inflection point:

Therefore, the coordinates of the inflection point =

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Repeat the investigation for three different values of A while maintaining b=4

Black = Orange = Blue =

−4 x

f ( x ) = Axe

, A = 5, 7, 20 (Working out in appendix)

The effect of changing the value of A on the graph y= Axe−bx when b remains constant is that if the value of A goes higher, the y value of the stationary point increases.

Black = Orange = Blue =

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SACE ID: 893239E The effect of changing the value of A on the graph f ( x ) = Ae−4 x (1−4 x ) when b remains constant is that when the value of A is higher, the y value of the stationary points decreases. Summary of results Stationary Point Coordinates

Point of Inflection Coordinates

1 10 ( , ) 4 4e 1 5 ( , ) 4 4e

1 20 ( , 2) 2 4e 1 10 ( , 2) 2 4e

A = 7, b = 4

1 7 ( , ) 4 4e

1 14 ( , 2) 2 4e

A = 20, b = 4

1 20 ( , ) 4 4e

1 40 ( , 2) 2 4e

A = 10, b =4 A = 5, b = 4

Through the table presented above, it is evident that A is directly proportional to the y coordinate of the stationary point of inflection. The x value in both the stationary point and the point of inflection remain constant throughout all the values of A meaning they have correlation to the b value which remains constant.

Repeat the investigation for three different values of b while maintaining A=10. −bx f ( x ) =10 xe

A = 10, b = 8, 13, 21 (Remaining workout in the appendix)

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Stationary Points: f’(x)=0Using x value: y coordinate =Therefore, the coordinates for the stationary points =

Inflection points for inflection point = when

Y coordinate:Therefo re, the coordinates for the inflection points =

Using a similar process, investigate the effect of changing the value of b on the graph of y= Axe−bx .

Black = Red = Blue =

When A remains constant and b is altered in f ( x ) =10 xe−bx , it is visible that when the value of b increases, both the x and y coordinates of the stationary point decreases.

Black = Red = Blue =

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When A remains constant and b is altered in f ( x )=10 e−bx (1−bx ) , it is visible that as the value of b increases, the x coordinate of each function gets smaller which makes the function steeper. Stationary Point Coordinates

Point of Inflection Coordinates

1 10 ( , ) 4 4e 1 10 ( , ) 8 8e

1 20 ( , 2) 2 4e 1 20 ( , 2) 4 8e

A = 10, b = 4 A = 10, b = 8

A = 10, b = 13

(

1 10 , ) 13 13 e

(

2 20 , ) 13 13 e 2

A = 10, b = 21

(

1 10 , ) 21 21 e

(

2 20 , ) 21 21 e 2

A conjecture on how the value b effects the x-coordinates of the stationary points and the point of inflection of the graph y= Axe−bx .

x=

Stationary point:

1 b

x=

Point of Inflection:

2 b

Proof of Conjecture:

y= Axe−bx

'

u= Ax u = A

v =e−bx v ' =−be−bx −bx

' ' Ae Product Rule=u v +u v −bx Ae (1−bx )

−bx

− Axbe

Stationary points = when f’(x) = 0

Ae−bx ( 1−bx )=0 −bx=−1

x=

−bx

Ae

≠0

1−bx= 0

1 b

Suitability of the surge function in modelling medicinal doses by relating the features of the graph to the effect that a medicinal dose has on the body.

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−bx f ' ( x )= Ae (1−bx ) −bx ' −bx u= Ae u =− Abe v =1−bx v ' =−b ' ' Product Rule=u v +u v −Ab e−bx + Ab 2 x e−bx− Ab e−bx ¿− Ab e−bx ( 2−bx )

Point of Inflection = when f’’(x)=0

¿− Ab e−bx ( 2−bx )=0 2−bx= 0

−Ab e−bx ≠ 0 2 −bx=−2 x= b

SACE ID: 893239E As seen, surge functions present a rapidly increasing peak followed by a slow decline. This is similar to how a human body responds to medicinal doses, a rapid increase is observed within the human body as the drug enters the body but this is soon diminished by the liver and vital organs. This shows the suitability of the surge functions in modelling medicinal doses. Limitations of the model 

  

As observed, the surge function will never reach zero completely, whereas the body will eventually remove all traces of the drug from the body, this means that a surge function can’t determine the moment when there is no of the drug left in the body. This model is on a normal human body, this does not account for different people and how they react to the certain drug, for some people the drug may be less effective due to weight and blood amount. Surge functions don’t account for factors that may alter the drugs effect in the body due to various circumstances, surge functions are smooth curves that show a slow decrease at the same pace. Surge functions don’t account for human activities and how they can affect the drugs effect, factors such as exercising can impact the effect of the drug on the human body.

Part 2: The logistic function A logistic function is in the form

P ( t )=

L 1+ A e−bt

where L, A and b are constants

and the independent variable t is usually time. Investigate the effect that the values of L, A and b have on the graph of the logistics function. Changing the value of L (A=10, b=4), L = 5, 8, 11 Purple = Orange = Green =

When the L value is altered and the A and b value remain constant, a direct correlation between the L value and the horizontal asymptote of the graph can be observed. The asymptote changes to the exact value of L.

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SACE ID: 893239E Changing the value of A (L=5, b=4), A = 5, 7, 20 Purple = Orange = Green =

By changing the value of A whilst the values of L and b remain constant, we can observe through the graph that, when the A value increases, the y intercept of the graph decreases.

Changing the values of b (L=5, A=10), b = 8, 13, 21 Purple = Orange = Green =

When the value of b in altered and the value of L and A remain constant, we can see that the higher the value of b, the closer the curve is to x=0. The y intercept does not change and the asymptote remains unchanged when we alter the value of b. Relate the specific features of the logistic graph to a limited growth model  

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An exponential growth model consists of one curve and increases to a certain limit whereas logistic graphs will increase to a limit and level off. Exponential growth models are used when growth is not limited by any resources meaning their growth can be very rapid, a logistic graph is more realistic as it limited by resources.

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An exponential growth model increases rapidly in scenarios such as child birth whereas a logistic growth models maintains a constant rate of growth until it reaches the maximum capacity.

Part 3: Modelling using Surge and Logistic Functions SCENARIO – The spread of a rumour in a classroom of 40 students. Through observation, when t=5, the rumour had spread to 10 students within the classroom. CONSTANTS – The limiting value for the number of people who will hear the rumour spreading is 40 since there are only 40 students within the classroom.

P (t )=

L 1+ A e−bt

where t = minutes after the student begins the rumour

(t ≥ 0)

. Finding the value of A. First, we must find the y intercept. When x=0

P ( 0 )= ¿

L 1+ A

L 1+ Ae−b ( 0 ) (because

0 e =1 )

The rumour begins with an initiator, the initial student who creates the rumour. So, when time = 0, only the initial student knew about the rumour. The max limit that the rumour can reach is the 40 students within the class. L = 40

P ( 0 )=

L 1+ A

1=

40 1+ A

1( 1+ A )=40

1+ A=40

A=39

Finding the value of b. We now know that the value of A = 39, L = 40 and that when t=5, the rumour will have spread to 10 students within the class.

P (t )=

9

L 1+ Ae−bt

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when t = 5, the rumour has reached 10 students.

10 ( 1+39 e

) =40

−b ( 5 )

ln b=

( 131 )

40 =1+39 e−b (5 ) 10

( )

3=39 e−b 5

40 ( ) 1+39 e−b 5 1 1 =e−b (5 ) ln =−5 b 13 13 10=

( )

b=.513

−5

Now we know that A = 39, b = .513, L = 40.

f ( x) =

40 −.513(x) 1+39 e

BLUE = GREEN =

DISCUSSION In this scenario, a logistic function has been used instead of a surge function because to it is more suitable to the situation. A surge function shows a rapid increase then a decline, in this scenario, the rumour can’t decline since it’s not possible for the students to just forget about it. The logistic function presented the slow start as only a few students knew the rumour and could spread it, but as more students learn about it, then each individual can help spread the rumour which is presented through the rapid increase. The increase slows down when there are only a few people left in the classroom of 40, ultimately levelling off at 40 since there are no more students for the rumour to be spread to.

LIMITATIONS

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

Not all students would want to indulge in spreading the rumour which is not taken into account in the graph. Some students would prefer to keep quiet even though they have heard the rumour, this could alter the rapid increase as it would take longer to reach the full 40 students. Depending on the situation within the classroom, the rumour might not be able to spread around as quick due to circumstances like it being a lesson where the students are having a test which prevents them from being able to communicate to each other. This means that the students would pass the rumour around very slowly or that it wouldn’t reach all the students.

CONCLUSION Within this investigation, both the surge function and the logistics functions and their use within real life applications were explored. For the surge functions, the conjectures that were found and proven on how the value of b effects the x coordinates of the stationary point and the point of inflection. For the logistic functions, we discovered that the value of L alters the horizontal asymptote of the graph, that the value of A alters the y intercepts of the graph and that the value of b affects the steepness of the graph as the higher the number, the closer it comes to the y axis. The scenario on the spread of a rumour within a classroom of 40 was presented to show the suitability of the logistic functions in these types of situation. Similarly, surge functions can be used in modelling medicinal doses as the surge model is very similar to the human body’s reaction to a drug. Both the surge and logistic models don’t consider the limitations and the situations that may occur within the experiment, aspects such as human nature and unexpected events are not taken into account which questions the appropriateness of the functions.

APPENDIX Repeat the investigation for three different values of A while maintaining b=4 A = 5,

−4 x

f ( x )=5 xe

Stationary Points: f’(x)=0Using x value: y coordinate =Therefore, the coordinates for the stationary points = 11

Inflection points for inflection point = when

Y coordinate:Therefo re, the coordinates for the inflection points =

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Now, A = 7,

Now, A = 20,

−4 x

f ( x )=7 xe

−4 x

f ( x )=20 xe

Using a similar process, investigate the effect of changing the value of b on the graph of y= Axe−bx . A = 10, b = 13.

Inflection points for inflection Stationary Points: point = when f’(x)=0Using x value: y coordinate =Therefore, the coordinates for the stationary points = 12

Stationar f’(x)=0Usin coordinate coordinate points =

Stationary Po ’(x)=0Using x coordinate =Th coordinates for stationary poin

Y coordinate:Therefore, the coordinates for the inflection points =

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Infl Stationary Points: po f’(x)=0Using x value: y coordinate =Therefore, the coordinates for the stationary points =

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