# Ib Math Portfolio Type 2

Topics: Function, Monotonic function, Convex function Pages: 25 (1557 words) Published: September 17, 2011
Introduction

In this task, I will develop model functions representing the tolerance of human beings to G-force over time.

In general, humans have a greater tolerance to forward acceleration than backward acceleration, since blood vessels in the retina appear more sensitive in the latter direction.

As we all know, the large acceleration is, the shorter time people can bear. Using the data shown in the task and Mat lab analysis, we can get several model functions to represent the tolerance of human beings to G-force over time. Then the model functions are improved through further investigation. The validation result shows that the models represented are efficient for the tolerance of human beings to G-force over time.

Problem formulation

To define appropriate variables and parameters, and identify any constraints for the data and use technology plot the data points on a graph. Comment on any apparent trends shown in the graph. Find function to model the behavior of the graph and explain the reason to choose the function. Create an equation to fit the graph. On new set axes, draw the model function and the function of the original data points. Comment on any differences and revise the model if necessary. Discuss the implications of the model in terms of G-force acting on human beings. Use technology to find another function to model the data. On a new set of axes, draw the model function and comment on the differences. Then, check whether the first model fits the new data. Make any changes to the model if it is needed. Discuss the limitation of the model and implication in terms of G-force.

Solving problem

Define the variables and parameters, we choose t as the time human beings could tolerant and g as the acceleration. The constraints of data are as follows:

Because t must be a positive number, when g is positive number it means a forward acceleration, if g is a negative number, it represent a backward acceleration.

Using Mat lab, we plot the data points as follows:

As time (t) increases, the tolerance of G-force (g) is getting smaller, and as time (t) decreases, the tolerance of G-force is getting larger. The trend would never have intercept with t axes and g axes. Because when t=0 and g=0, the time equals zero, the case would never happened, and when g equals zero, the time would tend to infinity, which means the g axes and t axes would never have intercept.

Through observation of the graph, the behavior of the curve is similar to that of inverse proportion function. As follows:

The definition of inverse proportion function is that, as x increases, the y decreases, and as x decreases, the y increases. The curve also has no intercept with x axes and y axes.

The statement of inverse proportion function is ,
So,

And the average number of k is,

Then, we plot the function and the data point in the same graph.

In the graph, it can be seen that, the curve of the function cannot fit the data very well. These two curves have the similar shapes, but the location is different. The curve of the real data is higher than the function, and some constant may exist between the function and real data. Thus, to describe the real data more exactly, we choose a constant and add it to the function above. We assume the new function from is So, we use the data give to solve parameter k and b, we choose first and fifth group of data, second and the sixth group of data, third and seventh group of data, fourth and eighth group of data to solve parameters separately. Then we get the average of them as the final parameters.

We also got the matrix’s form for equation sets as follows:

To get k and b, I use the method of solving matrix which is:

Thus,

We plot the graph with new function as follows:

Then we compare new function with two curves mentioned above in one graph as follows:

From the graph, it can be concluded that the new function can fit the data, and better than the first...

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