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Hi folks, I am seriously confused as to what can be a 7-pointer IA topic. Initially I wanted to do something related to Monte Carlo simulation like Risk assessment of an Investment portfolio but the problem is I am not able to show any mathematical calculations and according to my teacher if I don't put Calculus or Graph Theory then she won't give me a 7. If any of you guys know what Monte Carlo is and can help me somehow put calc in it I would highly appreciate that. Also if you have any other interesting topics that use Calculus or Graph theory please let me know. TIA!
Sure! here is an example- though do ensure that you clearly explain the utilization of calculus and graph theory concepts, document your methodology, and provide a comprehensive analysis in your internal assessment report. You would be surprised how many don't explain it clearly and get lower points:/

Estimating the value of pi using Monte Carlo Simulation

In this topic, calculus can be applied to estimate the value of pi by using the Monte Carlo simulation technique. The simulation involves randomly generating points within a square and determining the ratio of points falling within a quarter of a circle to the total number of points. The area of the quarter circle can be approximated by integrating the equation of the circle.
To incorporate graph theory, consider representing the simulation as a graph. Each point generated in the simulation can be considered as a node, and the distance between two points can be represented as an edge. Analyse the connectivity and shortest paths between nodes to determine the proximity of points within the quarter circle.
Then, formulate a mathematical model to describe the simulation. Define a function that represents the equation of the circle and use calculus to derive the area of the quarter circle as an integral of this function. Express the model in terms of the random generation of points and the calculation of the ratio of points falling within the quarter circle.
Implement the Monte Carlo simulation algorithm to generate a large number of random points within the square. Calculate the ratio of points falling within the quarter circle and use this ratio to estimate the area of the quarter circle. Multiply this area by 4 to estimate the value of pi.
Interpret the simulation results by comparing the estimated value of pi with the known value of pi. Analyse the accuracy of the estimation based on the number of simulated points used and discuss the convergence of the estimated value of pi as the number of points increases. Consider the limitations of the simulation and the impact of random variation on the accuracy of the estimation.
Reflect on the mathematical processes involved, such as the selection of the appropriate mathematical model, the accuracy of the approximation, and the convergence of the estimation. Discuss the implications of the findings and consider possible improvements to the simulation, such as using more sophisticated integration techniques or exploring other Monte Carlo simulation variations.

Feel free to adapt this to suit your interests, but hope this helps as a guideline :smile:

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