# Ib Math Portfolio Sl

Topics: Logarithm, Natural logarithm, Derivative Pages: 8 (1575 words) Published: November 15, 2012
Taipei European SchoolMath Portfolio
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VINCENT CHEN|
Gold Medal Heights
Aim: To consider the winning height for the men’s high jump in the Olympic games

Years| 1932| 1936| 1948| 1952| 1956| 1960| 1964| 1968| 1972| 1976| 1980| Height (cm)| 197| 203| 198| 204| 212| 216| 218| 224| 223| 225| 236|

Height (cm)
Height (cm)
As shown from the table above, showing the height achieved by the gold medalists at various Olympic games, the Olympic games were not held in 1940 and 1944 due to World War II. Year (1=1932, 2 = 1936 and so on)

Year (1=1932, 2 = 1936 and so on)

Using autograph, the graph above is a scatter graph showing the high jump results from the table.

The plot suggests that the high jump heights start off with a steep positive slope then coming to a decreasing negative slope, however without the 1940 and 1944 high jump competitions, it may not be certain. Then finally, it starts increasing again with a fairly steep positive slope.

However, it would not be realistic if the function has an infinitely increasing range, such as quadratic, exponential and linear because of the limitations that humans have due to natural forces like gravity. Therefore, narrowing down the options that may fit this graph to natural logarithm and logistics

Since the statistics given starts from year 1896, in order to make sure that calculations can be as simplified as possible, I have decided to rearrange the table with the assumption that 1896 is 0.

Years| 36| 40| 52| 56| 60| 64| 68| 72| 76| 80| 84| Height (cm)| 197| 203| 198| 204| 212| 216| 218| 224| 223| 225| 236|

This table shows the rearranged data of winning height for the men’s high jump in the Olympic games

High Jump Height vs. Years after 1896
High Jump Height vs. Years after 1896

Height (cm)
Height (cm)

Years after 1896
Years after 1896

With the natural logarithm equation y=a+b (lnx), the variables “a” and “b” are the unknown parameters that affects the shape of the curve. “a” affects the shift along the y-axis and “b” affects the vertical stretch of the line. “y” is the height and “x” is the year in centimeters.

In order to solve for the unknown, in this case “a” and “b”, two sets of given data points are substituted in the equation. (36,197) (84,236)

y=33.1+46(lnx)
Height (cm)
Height (cm)
High Jump Height vs. Years after 1896
High Jump Height vs. Years after 1896

Years after 1896

The graph above shows that the natural logarithmic function that I worked out doesn’t really fit the graph. y=33.1+46(lnx) Therefore, by finding the average difference in height compared with the plots can make this equation more accurate. (Average difference = 7.63cm) y=33.1+46(lnx) By subtracting the variable “a” by 7.63, it should give a more precise function that fits the graph. y=25.5+46(lnx)

High Jump Height vs. Years after 1896
High Jump Height vs. Years after 1896

Height (cm)
Height (cm)

Years after 1896
Years after 1896

Blue line = y=33.1+46lnx
Purple line = y=25.5+46(lnx)

From the new function y=25.5+46(lnx), it is clear that it fits the data plots more accurately.

By using technology, the TI-84 calculator, it is able to calculate a more precise equation for natural logarithm.

y=44+41lnx
High Jump Height vs. Years after 1896
High Jump Height vs. Years after 1896

Height (cm)
Height (cm)

Years after 1896
Years after 1896

Blue line = y=33.1+46lnx
Purple line = y=25.5+46(lnx)
Green line = y=44+41lnx

By using the natural logarithm function, it gives a more realistic prediction to the future since it shows a more constant positive slope that isn’t that steep, suggesting the fact that as human beings and the limitations from natural forces such as gravity, it is unlikely for athletes to continue jumping higher and higher in the future. There is a limit to how high athletes can jump, therefore the natural...