Main objective of the assessment: The objective of this task is to write a report on numerical approximation methods (maximum five pages long), using LaTeX and following the guidelines below.  In this task, wherever the parameters a and b appear, you should replace them with the last (a) and next to last (b) nonzero digits of your student number (e.g., if your student number is 1234567 then you should take a=7 and b=6 wherever a and b appear below; if your student number is 1203040 then you should take a=4 and b=3; etc.).

Description of the Assessment:  Each student must submit (as a .pdf file) a report, written using LaTeX (article style).  The maximum length of the report is 5 pages.  This report should be labelled with your student number, should include at least one table of results, at least one figure, at least one reference to a suitable resource (e.g., a book), and should also include the following:

1.   Title:  Choose a title appropriate to the content of the report.

2.   Introduction:  In this section you should introduce the report, and explain what you are going to do in it.

3.   Example 1:  application of polynomial approximation:  In this section, you should answer the following question:  The return on investment in a particular stock is assumed to have a   normal distribution, with mean zero and standard deviation 1.  The probability that the return on investment lies between α and β is then given by

1    ∫ β e −2 dx

Showing your working, derive an appropriate Taylor series expansion for e −2 , and hence

derive an approximation to the probability that your return lies between −a/(10b) and a/(10b).  Comment on the accuracy of your approximation.

4.   Example 2: application of root finding techniques:  In this section, you should answer the   following question:  Terry the Tortoise and Harry the Hare are having a race.  Both start running at the same time (t = 0).  Terry runs at a steady pace along the course, so his position at time t is given by:

T = t.

Harry gives Terry a head start, and runs more erratically, sometimes going forward and sometimes going backward, with his position at time t (relative to Terry’s starting position) given by:

H = t3 − (1900b + 260a) t2 + (1 + 100a (13a + 380b)) t − 10000a2 b

Using one or more appropriate numerical methods, find an approximation to the value of each of the times at which Terry and Harry are at the same location, explaining clearly which method you have used and showing your working.  Explain in words (using a plot to assist your explanation) how the race progresses.

5.   Conclusion:  In this section, you should summarise your report, and draw some conclusions   about how numerical approximation methods may be useful in different areas of mathematics, drawing links in this discussion to relevant material from other modules on your degree course.

6.   References:  Include a reference to at least one suitable resource (e.g., a book), which should be cited at the appropriate location in the main text. This reference should be added to the bibliography using BibTeX.

Learning outcomes to be assessed: The module learning outcomes relevant to this assessment are:

•   Plan and implement numerical methods using an appropriate programming language. Illustrate the results using the language's graphicsfacilities. Analyse and interpret the results of the numerical implementation in terms of the original problem;

•   Demonstrate the knowledge and understanding of the multiple skills necessary to operate in a professional environment

Marking: the total mark available for this assignment is worth up to 25% of the available overall mark for the module.  Marks (out of 100) will be allocated as follows:

•    30% of the available marks will be for the quality of the mathematics and the explanations related to Example 1.

•    30% of the available marks will be for the quality of the mathematics and the explanations related to Example 2.

•    40% of the available marks will be for the quality of the LaTeX, presentation, and writing, and for following instructions as provided in this assignment brief.

Submission instructions:  Submission should be through WISEflow.  Each student should submit two files:

1.   A single .pdf file, containing the report.  The name of this file should include the module code and your student ID number, e.g. MA2690_1234567.pdf.

2.   A zip file containing the source .tex file, .bib file, and any other files (e.g. figure files) used to generate the .pdf.  The name of this file should also include the module code and your student ID number, e.g. MA2690_1234567.zip.

If you are unsure how to download your files from Overleaf into a folder on your computer, then please follow the instructions given in the following link:

https://www.overleaf.com/learn/how-to/Downloading_a_Project

Note that the first part of the instructions creates a .zip file containing all of the source files but not the .pdf file.  You will need to download the .pdf file separately by following the instructions on how to download the finished .pdf.  Please remember to back up your files periodically; it is your responsibility to make sure that your files are securely backed up, and the safest way to do this is by    using the filestore at Brunel – details of how to do this can be found at: