• Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

Mathematical Modeling

Extracts from this document...


Year 12 Mathematics Standard Level

Mr. A. Mumm

Medora Choi

8th March, 2010

Mathematics SL Portfolio: Type II

China’s Population Growth

        Mathematics is a study of the concepts of quantity, structure, space and change. It is a type of science that draws conclusions and connections to the world around us. Mathematicians would call math a science of patterns and these patterns are discovered in numbers, space, science, computers, imaginary abstractions, and everywhere else. Mathematics is also found in numerous natural phenomena’s that occurs around us. Today math is used all around us and is applied to many educational fields, through this people have become inspired to discover and make use of their mathematical knowledge which will then lead to entirely new disciplines. Math is present in wherever there are difficult problems that involve quantity, structure, space or change; such problems appear in various forms such as commerce, land measurement and especially astronomy.

The purpose of this paper is to examine the different functions and investigate the best model for the population of China from 1950 to 1995. The following table shows the population in million of China during the year of 1950 to 1995:












Population in Million











...read more.


To test out this function, I have to choose two points from the data to find out the gradient of the function,

First Point: (0, 554.8)

Second Point: (5, 609)

To find the gradient, I use the formula


Therefore it is shown as:

m =image18.png

The next step is to find b in the function, I can find this value by inputting a data point into the function like this:


Thus, generating the complete linear function:


I then input this function into the graph to test the accuracy of the hand-calculated function.


As seen from the graph above, the line does not show the best fit as it does not touch upon all the lines, suggesting that the hand-calculated function may not be entirely reliable.

I then used my GDC to find the value for m and b, which is:


I then plot it into GeoGebra and it fits the line perfectly as shown as below:


To find a better fit line for the graph, I used a [FitLine] function from the GeoGebra to find a more accurate function, which looks like this:


This shows that the GDC and the GeoGebra are much more reliable resources than hand-calculated.

...read more.


         To further compare the two models, there are additional data on population trends in China from the 2008 World Economic Outlook, published by the International Monetary Fund (IMF) which is shown on the table below:









Population in Millions








The above data is individually tested on each of the models. For the first model, which is also the linear model, has only three points touching the model, including year 1983, 1992 and 1997; the rest of the data points were lower than the model. The second model is then tested, and for this researcher’s model, all the points are extremely close are touching the model, meaning the modified model should be based on the researcher’s model as it is more reliable than the linear model.

Hence, this model is established:


This model fit all the IMF data; hence it applies to all the given data from year 1950 to 2008 where the maximum points can touch on this model as shown below:


The modified model shows that the general data points are on the line, meaning this is the best that fits all the data since 1950 until 2008.

...read more.

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

Found what you're looking for?

  • Start learning 29% faster today
  • 150,000+ documents available
  • Just £6.99 a month

Not the one? Search for your essay title...
  • Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

See related essaysSee related essays

Related International Baccalaureate Maths essays

  1. Extended Essay- Math

    ���L��(tm)����ֲٲ�_�"�<��Ý�'W'"���T +X*d+t),+J*�)V*>Q")(tm)*e(u(-)K) (�R�PaR�Q9 ҧ�(c)���zQu^MX�G��3ufu;��� C� �k���q�W4��'� �jÔ�� �(r)Õ�ԡ��(tm)����=�;�' G�"�{�/��_�?k ab�d��P�0ư�p�H�(ͨ�elf|�x�"l�lRa��T�4���(tm)�Y�Y�9�����3 - ?��EK5�4�~+Z+G" "w�'�1�]6���M'�K[��]`-�(r)�]�����(r)�c���+�?8(8�u������������Ӥ��s1/4s� 1/2��K�˪"�k��"��[��wNw�{� ��ţ�ce���"g<U<s=�1/2�1/4'1/4������71/27����է�g���RMY��=�"�g�W��_߿�>@' 0`6P'�0p.H'�(h>X/�$x�jD� ....��� Y �z>t;�5�%��A����L�|%�5�}"z1�*�.����cF'áx��ot*~$�$^MbL�HJ-L�K�M1M9-�N�K��+�7k��4����P�oz_�PFN�L�Yf}MVh��l����o�\�u���d�L�7�!-.7&�����R �)��m�?t?_>�$���G�"�.,>�z��1ì±c����2�N���*�v��1/2�'Ó¥4��Se�e�����7*�+�VV���>(tm)wr�ʿj�"�(c)��yN�y�zf��ٶj��'lMBÍZ-�"���5�q���m��8?U�P�� �����x�|!��|"g�H�qs�E(tm)�g[X[�/�K�->^��<~��J�U�"��\;���z� jKn[ln��p�x�i��ץ��z]���7*o��<�MÓ�1/2}+��JOT�BoP�t�w��m��O�� ��޹}���A���4�u�W��@�AÛ�P�C���ê�m��-u�h�t=�~�=�7�;f<v��" Om�>-w-�x�lj�b�y�� /�'3_b^-z���5���7oZ�T�n3/45~;��������7fr>�>"���6�)�Ý7�����̧�O� ��?��"���_� -�-�,�,m/���7�o}+v+���__=��G����Ý(r)?g�7pe��][V[/�÷��(1"_� i�@-�#5";R;�@C�"��Å"+Â` !�-6'��a� �A�0 ��Ө�2:#�(tm)�V�B��Q:"L�"1�I�[0�0�%�b�e�e1/2Â���������]�x �)|N"Y,S|V�V�E�N�O���-R��i�~Õ·jk���Z'��:ƺ�z-�� ��F%�&]���^��Y�Z��l�m�v)�i�:X8�:98;"�����"{xx�����������@��5��W � � b ��S�"1/4 }vY�""��D'�Pb �8�3/4�&"&F&Y& %o�<Km�{0�']-� Y[׳ �����0�����-:|P;�5o3}X�H�Q�cW�o�K��-'/U(S*W(r)P(c)T9(c)Z�y��t�(tm)��5,��1/4�"Χ��6o�1/4p(r)(c)�����-�K��\��6�&�-�Q����ì­n�[ =.1/2...}s���w-�};�x�@d��p죰��j�1/4c4ckO��>���k���[/z'"_�1/4:�:������'w���+g�}�?�<>����"�y���;_j��] Y�]V�*�Mb��{����o686ݶj��w�'%:��>@bP$�s���2* �}#����dq_�}"*�4�?��'Î�-!�1�\�4��U�Í1/2���Û�0�0?I�^�ÐQ�D,A1/4N��7iqYe9 y %E %^eH�� �[ jtj6hUh�����ַ504T3'56�4e0Ú��/Z�XNX

  2. Maths Portfolio - Population trends in China

    *[plotted on GeoGebra4] We can see that the linear model did not work out as accurate as with the first model for the first table of data. The possibilities for that are because this has some newer data points that don't increase with the same intensity as before.

  1. IB Mathematics Portfolio - Modeling the amount of a drug in the bloodstream

    Also, in the data given there is a little minor mistake. Between the 7.5th hr to the 8th hour after the initial dose of drug, the amount in the blood stream remains the same. I thought it might be a human error because I didn't think the amount of drug would stay the same for that specific hour.

  2. A logistic model

    The interval of calculation is 1 year. Please note that from year 6 and onwards the population resonates above and below the limit (yet due to the rounding up of numbers this is not observed), and it finally stabilizes in year 17 (and onwards) where the population is exactly equal to the sustainable limit.

  1. Tide Modeling

    All of the vales above can also be used for sine since cosine and sine are only a horizontal translation of each other. No transformation would be required because as the graph is half way up it crosses the y- axis and it also happens in the transformed.

  2. Finding Functions to Model Population trends in China

    more points than linear function, such as coordinates from year 1970 to 1990 have all failed to meet the line of the best fit as they tend to be more inclined, compare to other coordinates. By using constant controller, I was able to visually understand how the changes of the parameters can determine the shape of the graph.

  1. Function that best models the population of China. Some of the functions that ...

    the exponential function: I will now find out the values for a and b. In order to find a and b, I will randomly pick for values for x and y. Then in both the equation I will isolate for a.

  2. Creating a logistic model

    un If we let 10000 fish with an initial growth rate of 2.3 cultivate for 20 years, the population of growth would be: Year Population of Fish Growth Rate t un+1 = (-0.000026 � un + 2.56) un rn = -0.000026 � un + 2.56 0 10000 2.3 1 23000

  • Over 160,000 pieces
    of student written work
  • Annotated by
    experienced teachers
  • Ideas and feedback to
    improve your own work