two matlab assignments

University of Derby School of Electronics, Computing and Mathematics In-course Assignment Specification

Module Code and Title: 4MA502 – Mathematical Software Assignment No. and Title: 1 – Coursework Part 2 Assessment Tutor: Sam O’Neill Weighting Towards Module Grade: 40 % Date Set: 03/04/2020 Submission Date: 11/05/2020 – 23:59

Penalty for Late Submission

Recognising that deadlines are an integral part of professional workplace practice; the University expects students to meet all agreed deadlines for submission of assessments. However, the University acknowledges that there may be circumstances which prevent students from meeting deadlines. There are now 3 distinct processes in place to deal with differing student circumstances. https://www.derby.ac.uk/about/academic-regulations/ Assessed Extended Deadline (AED) Students with disabilities or long term health issues are entitled to a Support Plan. Exceptional Extenuating Circumstances (EEC) The EEC policy applies to situations where serious, unforeseen circumstances prevent the student from completing the assignment on time or to the normal standard. Late Submission Requests for late submission (LSR) will be made to the relevant Subject Manager in the School (or Head of Joint Honours for joint honours students) who can authorise an extension of up to a maximum of one week.

Level of Collaboration:

This is an individual assignment. No collaboration with other students or anyone else is allowed.

Learning Outcome(s) covered in this assignment: Demonstrate a knowledge of the underlying concepts of optimisation and metaheuristics and apply appropriate techniques to a range of problems; Derive solutions to various problems through the application and implementation of modern optimisation techniques.

Submissions in Turnitin and Blackboard You must submit your work using your student number to identify yourself, not your name. You must not use your name in the text of the work at any point. When you submit your work in Turnitin you must submit your student number within the assignment document and in the Submission title field in Turnitin.

Marking Criteria: Credit will be awarded for:

70–100%: Excellent

Outstanding; high to very high standard; a high level of critical analysis and evaluation; commendable originality; high quality presentation; exceptional clarity of ideas; excellent coherence and logic.

60-69%: Very good

A very good standard; a very good critical analysis and evaluation; significant originality; well researched; a very good standard of presentation; pleasing clarity of ideas; thoughtful and effective presentation; very good sense of coherence and logic;

50-59%: Good

A good standard; a fairly good level of critical analysis and evaluation; some evidence of original thinking; quite well researched; a good standard of presentation; ideas generally clear and coherent, some misunderstandings; some deficiencies in presentation.

40-49%: Satisfactory

A sound standard of work; a fair level of critical analysis and evaluation; little evidence of original thinking or originality; adequately researched; a sound standard of presentation; ideas fairly clear and coherent, some significant misunderstandings and errors; some weakness in style or presentation but satisfactory overall.

35-39%: Unsatisfactory (marginal fail)

Overall marginally unsatisfactory; some sound aspects but some of the following weaknesses are evident: inadequate critical analysis and evaluation; little evidence of originality; not well researched; standard of presentation unacceptable; ideas unclear and incoherent; some significant errors and misunderstandings.

1-34%: Very poor (fail)

Well below the pass standard; a poor critical analysis and evaluation; no evidence of originality; poorly researched; standard of presentation totally unacceptable; ideas confused and incoherent, some serious misunderstandings and errors. A clear fail, well short of the pass standard...The work demonstrates nothing of merit.

IMPORTANT GUIDELINES FOR SUBMISSION

All 4 questions must be answered using either Excel or MATLAB. You are encouraged to use other methods/means to verify your Excel/MATLAB solutions.

You are required to submit a report containing a write up of your answers/results and submit any Excel or MATLAB files that were used to in answering the questions.

This is an individual assignment. No collaboration with other students or anyone else is allowed. Please be aware that this is obvious when MATLAB/Excel files and report answers are extremely similar. You MUST not collude with other students, students expected of collusion may be penalised subject to the academic regulations

Question 1 Derive and implement a Forward Differences Scheme to numerically solve the following and provide a plot of your solution between 𝑥 = 0 and 𝑥 = 5.

Use a step size ℎ = 0.1, additional marks can be gained through analysis (e.g. errors, changes in step size etc…)

a) = 4𝑦, 𝑦(0) = 6

(15 marks)

b) = 𝑥𝑐𝑜𝑠(𝑥) + 2, 𝑦(0) = 8

(15 marks)

Question 2 Use Newton’s Method to solve the following to a tolerance of 0.001 using the given starting points 𝑥 . Provide a table of results and a plot of the iterations. Investigate whether the choice of 𝑥 matters?

i.e. successive terms should differ by less than or equal to 0.001. i.e. |𝑥 − 𝑥 | ≤ 0.001

a) 𝑥 + 3𝑥 + 4𝑥 + 3 = 0, 𝑥 = 1 (10 marks)

b) 3𝑥 + 3 = 𝑒 , 𝑥 = 3 (10 marks)

Question 3 Use numerical integration to evaluate the following, provide a table to summarise your results. How could you improve the evaluations?

a) ∫ sin , use 10 intervals, 𝑁 = 10

(10 marks)

b) ∫ √1 + 𝑥 , use 8 intervals, 𝑁 = 8 (10 marks)

Question 4 Answer the following problems:

a) The product of the first five natural numbers is,

1 ⋅ 2 ⋅ 3 ⋅ 4 ⋅ 5 = 5! = 120

The square of the sum of the first five natural numbers is,

(1 + 2 + 3 + 4 + 5) = 15 = 225

Hence the difference between the product of the first ten natural numbers and the square of the sum is 120 − 225 = −105.

Find the difference between the product of the first fifty natural numbers and the square of the sum of the first fifty natural numbers.

(6 marks)

b) By listing the first six prime numbers: 2, 3, 5, 7, 11, and 13, we can see that the 6th prime is 13. What is the 2021st prime number? Hint, a useful algorithm to look to implement is the Sieve of Eratosthenes

(12 marks)

c) You have the chance to play the following game. Each round cost £4 and you win £𝑛 per round, where 𝑛 is your score for that round. You can play as many rounds as you like. Each round is as follows - You roll a dice. If the dice lands on 3 then you can roll again, if it lands on anything else then you stop. Your score for the round is the total of all your rolls in that round. e.g. First round Roll 5, STOP Round Score 5 Second round Roll 3, Roll 3, Roll 1, STOP Round Score 3+3+1 = 7 Third round Roll 1, STOP Round Score 1 Total score after 3 rounds of 5+7+1 = 13 Therefore, you made a profit of £1 in total, as you played 3 rounds costing £12, but won back £13. a) Given that you have plenty of money to play with, would you keep playing the game? Use a simulation

to decide. b) If the game is changed so that you can continue to roll on a 1 instead of a 3. Would this affect the

choice of whether you would play? (12 marks)

Total 100 marks