Describe how and why one might approximate a function by a Taylor polynomial.

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

2.Polynomial approximation: In this section, you should answer the following questions. This section should include at least one table of results, at least one figure, and at least one reference to the wider literature:

•Describe how and why one might approximate a function by a Taylor polynomial;

•As an example, determine a Taylor polynomial approximating ln|cos(𝑥𝑥)| near 𝑥𝑥=0and containing at least three non-zero terms;•Show via appropriate plots how the accuracy of your Taylor polynomial relates to the degree of the polynomial, and to the value of 𝑥𝑥;

•Explain how your polynomial approximation could be used to calculate an approximation to the value of:� −ln |cos (𝑥𝑥)|𝑐𝑐−𝑐𝑐𝑑𝑑𝑥𝑥,where𝑐𝑐=(25 +𝑚𝑚)𝜋𝜋90, with 𝑚𝑚 the last digit of your student number, and how the accuracy of this approximation depends on the degree of the polynomial. Illustrate your findings using an appropriate table of results.

3.Finding roots to nonlinear equations: In this section, you should answer the following questions. This section should include at least one table of results, at least one figure, and at least one reference to the wider literature:•Describe how you would find all solutions to the equation:𝑥𝑥3+𝑎𝑎𝑥𝑥2−𝑥𝑥+𝑏𝑏=0,where 𝑎𝑎 and 𝑏𝑏 are the next to last and the last nonzero digits of your student number, using:i.The bisection method;ii.Fixed point iteration;iii.Newton’s method.•To help you choose a suitable interval or initial guess, plot the function using MATLAB.•Explain the steps of each method, discuss the advantages and disadvantages of each scheme, present a table of results for each approach and explain your findings.
4.Conclusion: In this section you should summarise your findings in the report5.References: You should list at least two references, which should be cited at the appropriate locations in the main text. These references should be added to the bibliography using BibTeX.6.Appendix: In the appendix you should include all MATLAB code used to generate the results in sections 2 and 3. There is no page limit for the appendix.