Maths SL

Portfolio Type IIPopulation trends in China

Selina Novak 8L

Portfolio 1: Population trends in China

In this assignment, I will investigate different functions that best model the population of China.

In Table 1, the population of China from 1950 to 1995 is given:

Table 1

Year 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995Population in Millions

554.8 609.0 657.5 729.2 830.7 927.8 998.9 1070.0 1155.3 1220.5

The population is given in millions. Since people can be counted, the table shows numerical values.Using this data to draw graphs, the years will be the x-values, while the number of people in millions will be the y-values. The Domain of the data is { }, since the years can be positive, negative an zero. For instance, there have been people living in China not only in A.D., but also in the year zero and in B.C. The Range of the Data is { } since the population in China was never zero or negative.

Using technology I plotted the data points from the above table on a graph:

Figure 1

In Figure 1 one can observe, that the population of China has grown rapidly over the specified period. However, the growth rate has not been constant. The population slightly fluctuated during its increase. Moreover, it can be seen on the graph, that there was a marked increase of population between the years 1965 and 1975. The reason for the slower increase of population after 1975 can be due to the fact that the one-child-policy has been installed.

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In order to find the types of functions which could best model the behaviour of the graph, I found the lines of best fit. In my point of view, the linear, as well as the exponential lines of best fit are most suitable to describe the growth of population in China.

Figure 2

The graph is nearly linear, as can be seen when adding a linear line of best fit (see Figure 2), however in real life situations like this the function is never completely linear, as the data slightly fluctuates. Moreover, the linear function moves into both the positive and the negative y-direction to infinity, which is not possible in real life. A negative number of people does not exist, and the earth does not have the capacity to hold an infinite number of people.As a consequence, a linear function would not represent the given data exactly, but simplify it.

Figure 3

An exponential line of best fit is even better (see Figure 3) than the linear approach, as the population grows in a slight slope. Moreover, with an exponential function, the x-axis constitutes an asymptote. Since the population in China was never zero or less, this approach fits very well.

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As a next step, I will analytically develop one model function that fits the data points on my graph. Having chosen the linear approach, I will set up a linear function.

A linear equation is P(t)=mt+c where m is the gradient of the function, t is the variable and c is a constant. For my model function I will draw a line through the outer most points.

Equation 1

Consequently, the model function can be described as the function P (population) depending on t (Time):Equation 2

Figure 4: My linear model function

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According to this graph (Figure 4), the population does not fluctuate. Moreover, the line increases and decreases to infinity. Obviously in real life this trend can not keep growing like that forever. In addition, the population of China was never zero or negative. Consequently, this line graph is only appropriate for a short period, such as from 1950 to 1995.Since the linear model is neither realistic nor very reliable, I will revise my model and set up an exponential model function which describes the given data in a better way.

An exponential function is where a and b are constants and t is a variable.

Having chosen the two outer most points, I will set up two equations:

Equation 3: Part I

Using the substitution method I will first find the unknown b and then a.Equation 3: Part II

By inserting my result for b into one of the equations, I will get a.

Equation 3: Part III

As a result, my exponential model function is defined by:Equation 3: Final Formula

132 ( ) 8.21551287 10 1.017665294tP t

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Figure 5: My exponential and linear model functions

Blue…my linear model function,

Pink…my exponential model function,

On this graph one can see, that my second model function, the exponential approach, fits the given data better. Even though both functions nearly go through the same amount of points, the shape of the data, which is a slight slope, is better described by the exponential model.

A researcher suggests that the population P at time t can be modelled by:

Equation 4

From my prior mathematical knowledge I can say that this is the formula of logistic growth. With the help of regression on my calculator I estimated and interpreted the values for K, L and M.

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GDC 1

Using these estimates, my GDC plotted the following graph:GDC 2

Interpreting these values I found out that, in this formula, K determines the stretching of the logistic function along the y-axis. When the value of K increases, the graph is stretched into the positive y-direction, while when K decreases, the function is shrunk into the negative y-direction.Looking at the denominator, L is responsible for the movement of the graph along the x-axis. As the value of L increases, the graph is moved into the positive x-direction, while as L decreases, the function is moved into the negative x-direction.Furthermore, M is responsible for the horizontal stretching of the function. Because in this formula there is a minus in front of the M, when M decreases, the function is stretched into the positive x-direction, whereas when M increases, the graph is shrunk into the negative x-direction.

Hence, I constructed the researcher’s model, using my estimates:

Equation 5

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Figure 6: Researcher’s model

Just like in the exponential function, in this graph the x-axis constitutes a horizontal asymptote. However, this model function is more realistic than the exponential function, since the population does not grow to infinity. Nonetheless, the researcher’s model suggests that from 2080 onwards the population will stay the same to infinity. This is not realistic, since there will always be a variation in data.Moreover, according to the researcher’s model there is a steady increase without any fluctuation of the population of China in the period under investigation. This is never the case in real life.

There are different implications concerning the growth of China’s population in the future given by my linear, my exponential and the researcher’s model functions.According to the linear model function, the growth rate of China’s population will remain as high as it has been since 1950. Moreover, the population is predicted to grow to infinity.Similarly, the exponential function, implies that the population of China will continue to grow rapidly until infinity. It is interesting no note that the growth rate in the exponential function will increase sharply.The researcher’s logistic model function, implies that the growth rate will decrease again

until approximately 2080. According to the population of China will stop growing in about 2080 and then remain the same to infinity.The implications of each of these models in terms of population growth for China in the future all, sooner or later, turn out to be incorrect and unrealistic. The most appropriate model is the researcher’s function, as it predicts a decrease in the growth rate of the population at some point.

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In Table 2 the population of China from 1983 to 2008 is given in millions:Table 2

Year 1983 1992 1997 2000 2003 2005 2008Population in Millions

554.8 609.0 657.5 729.2 830.7 927.8 998.9

In the following step I will draw a graph with the data from 1983 to 2008 and comment on how well the researcher’s model and my model functions fit the data.

Figure 7: Population 1983-2008 Researcher's + My Models

Blue… my linear model function,

Green… my exponential model function,

Red… researcher’s model (logistic function),

From this graph one can clearly see that the exponential function does not fit the given data at all. While the exponential function rapidly increases, the growth of the population steadily decreases.Until 1997 both the researcher’s model and my linear function describe the given data very well. After that, there has been a drop in the population growth rate which has not been expected by both models. However, the researcher’s model predicts a decline in the growth rate at some point, so the shape of his model is more accurate than the linear approach.

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As a next step, I chose the researcher’s model to be the best fitting one and modified it

into , so that it applies to all the given data from 1950 to 2008.

Equation 6

Figure 8: Researcher’s model + modified Resarcher’s Model

Red…Researcher’s model,

Green…My modified researcher’s model,

In this formula, K is responsible for the stretching of the logistic function along the y-axis. As the value of K increases, the graph is stretched into the positive y-direction and as K decreases, the function is shrunk into the negative y-direction. With regard to this graph, we need to shrink the function so that it fits the given data from 1983 to 2008. Consequently, I changed the initial value of K ( ) into 1617.

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In the denominator, L is responsible for the movement of the function along the x-axis. As L increases, the function is moved into the positive x-direction, while as L decreases, the graph is moved into the negative x-direction. I changed the initial L ( ) into

since this way the graph is moved into the positive x-direction. Moreover, the value of M is responsible for the horizontal stretching of the graph. Since in the formula there is a minus in front of the M, when M decreases, the function is stretched into the positive x-direction, while when M increases, the graph is shrunk into the negative x-direction. In this special case, I changed the initial M to 0.03993. Consequently, the logistic function is shrunk into the negative x-direction.The modified model fits the data better than the initial researcher’s model. While the researcher’s original function estimates, that the population will continue to grow as rapidly as it did since 1950, my modified function shows that the growth rate of China’s population decreases. However, the modified model implies that from about 2060 onwards the

population in China will remain the same to infinity, while in the population keeps on growing until approximately 2100. Nonetheless, in both functions the population will stop growing at some point and then stay the same to infinity, which is not realistic at all, since the number of people living in China will always vary.Furthermore, with a logistic function the x-axis constitutes a horizontal asymptote, which is appropriate for this interpretation of data, since the population in China was never zero or negative. Nevertheless, in neither one of the both models the population growth fluctuates as it does in real life. As a consequence, my transformed researcher’s model is more appropriate to describe the given data, however it simplifies it to a certain extent as well.

In conclusion, in this task I investigated different functions which best model China’s population growth.At first I plotted the given data from 1950-1995 and found an appropriate model function. As a linear function was not fully appropriate, I developed an exponential function.As a next step, I found my estimates for the researcher’s model and compared it with my own models.In the following section of the task, I used all functions to describe the population growth from 1950 to 2008 and chose the one which could describe the new data best.In the last part of the portfolio, I transformed the best fitting model, so that it applied to all the given data from 1950 to 2008.

I presume the linear line of best fit would be more applicable, when the heights of gold medalists after 1948 are only considered. Figure III

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Linear Line of best fit for Height of Gold medalists(1952-1980)

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1950 1955 1960 1965 1970 1975 1980 1985

Years Of Various Olym pic Gam es

Hei

gh

ts O

f G

old

Med

alis

ts(c

m)

Gold medalistsheight

Linear (Goldmedalists height)

Figure III is a graph showing the original graph of the height of gold medalists and its linear line of best fit after 1948. On the graph, the linear line of best fit is at its best since it passes through the center of the original graph ( see figure III). The linear function moves into both the positive and the negative y-direction equally. However

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