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Function that best models the population of China. Some of the functions that I think that could model this data were linear function () and exponential function ().

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Introduction

Internal Assessment 2

Function that best models the population of China

This is the data of China’s population from 1950 to 1995:

Time (Year)

Population         (in millions)

1950

554.8

1955

609.0

1960

657.5

1965

729.2

1970

830.7

1975

927.8

1980

998.9

1985

1070.0

1990

1155.3

1995

1220.5

In this given data, the x value is the time (year) because time is always independent. Whereas the population (in millions) of china is the y value because each value is dependent on the time. Time cannot be stopped and so the values of x are from 0 years to infinite years. The y value that is the population is limited (image00.pngimage00.png) because of numerous of reason, for example less food availability we cannot have infinite people..

Now, I am going to plot this given data from 1950 to 1995 using Logger Pro. Steps are shown in the appendix:

image10.png

Graph 1

The reason why there is not a line connecting all the points is that because it might be possible that in 1951 or 1969 or in another period there could have been a rapid growth or decline in the population. That is why it is not continuous but discrete.

As we can see that the population is increasing as by the time. As the years are passing by, the trend of the population is gradually increasing more and more, the graph is becoming steeper.

...read more.

Middle

y = 554.8

x = 1950

y =1155.3

x = 1990

b = 1.018506859

image21.png

b = 1.018506859

image21.png

Out of all this, I decided to choose image22.pngimage22.png, using the years 1950 and 1990. I will now graph it using TI-84 Plus:

image23.png

Graph 3

This model is the best fit to the data. But as we can see that some of the points are quite off. From above this is the best that I could see.

A researcher suggests that the population, image24.pngimage24.png at time image25.pngimage25.png can be modelled by

image27.png

Where image28.pngimage28.png, image29.pngimage29.png, and image30.pngimage30.pngare parameters.

This is a logistic equation and the parameters in this equation each have a special role to perform:

Parameters

Role

image28.png

When the value of image28.pngimage28.png is increased the graph is shifted up by a certain number. This depends on the equation itself by how much the graph has shifted. On the other hand, when the value of image28.pngimage28.png is decreased the graph is shifted down by the same amount. We can also say that the value of image28.pngimage28.png is responsible for the up and down of the horizontal asymptote.

image29.png

When the value of image29.pngimage29.png increases, the x shifts towards right and when image29.pngimage29.png decreases, the x shifts towards left. Or in other words, the value of image29.pngimage29.png is responsible for the shift left and right of the vertical asymptote.

image30.png

When the sign of the image30.pngimage30.png

...read more.

Conclusion

Time (Year)

Population         (in millions)

1950

554.8

1955

609.0

1960

657.5

1965

729.2

1970

830.7

1975

927.8

1980

998.9

1985

1070.0

1990

1155.3

1995

1220.5

1983

1030.1

1992

1171.7

1997

1236.3

2000

1267.4

2003

1292.3

2005

1307.6

2008

1327.7

image46.png

Graph 8

In order to check it, I will use my TI-84 Plus. Method is shown in the appendix. I will find the parameters of the researcher`s model first and then graph it with my calculator. This is what my TI-84 Plus showed me:

image47.png

Therefore,

image48.png

image49.png

Now I will use this new equation to graph again with my TI-84 Plus. Method is shown in the appendix:

image50.png

Graph 9

As we can see that the graph from my TI-84 Plus matches the graph that I got with the help of Logger Pro. This new modified version of the model fits the data from 1950 to 2008 fits very well as it almost goes through each point. Also there is no big jump in this particular graph. The graph almost looks like it is easily making its way up.

 Also, this researcher’s model is not always going to work. If there are earthquakes or natural disasters the population will decline. Also it will take time to get back in track. Declining of population is possible but not so common. For example,

image51.png

Graph 9

The graph will not be able to cover all the points. In short, other graphs should also be used because this graph will not always work, for example when the population actually starts declining.

...read more.

This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.

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