Mathematical Technology and Thinking

Mathematical Technology and Thinking
coursework deadline
Learning Outcomes:
A) Use software such as spreadsheets to tackle mathematical problems.
Plagiarism is presenting somebody else’s work as your own. It includes:
copying information directly from the Web or books without referencing the
material; submitting joint coursework as an individual effort; copying
another student’s coursework; stealing coursework from another student
and submitting it as your own work. Suspected plagiarism will be
investigated and if found to have occurred will be dealt with according to the
procedures set down by the University. Please see your student
handbook for further details of what is / isn’t plagiarism. Details are also on
the Student Intranet.
All material copied or amended from any source (e.g. internet, books) must be
referenced correctly according to the reference style you are using.
Your work will be submitted for plagiarism checking. Any attempt to bypass our
plagiarism detection systems will be treated as a severe Assessment Offence.
Coursework Submission Requirements
• An electronic copy of your work for this coursework must be
fully uploaded by midnight on the Deadline Date
of Wednesday 17/02/2016 using the link on the coursework
Teachmat page for MATH1103.
• For this coursework you must submit a single Acrobat PDF
document.
In general, any text in the document must not be an image (ie
must not be scanned) and would normally be generated from
other documents (eg MS Office using “Save As .. PDF”). More
details are on the IT Support pages . An exception to this is
hand written mathematical notation, but when scanning do
ensure the file size is not excessive.
• There are limits on the file size (current values are on TeachMat
and the Student Intranet).
• Make sure that any files you upload are virus-free and not
protected by a password or corrupted otherwise they will be
treated as null submissions.
• Your work will not be printed in colour. Please ensure that any
pages with colour are acceptable when printed in Black and
White.
• You must NOT submit a paper copy of this coursework, or
include the Banner header sheet.
• All courseworks must be submitted as above. Under no
circumstances can they be accepted by academic staff
The University website has details of the current Coursework Regulations,
including details of penalties for late submission, procedures for
Extenuating Circumstances, and penalties for Assessment
Offences. See http://www2.gre.ac.uk/current-students/regs
Assignment specification
THIS COURSEWORK WILL BE MARKED ANONYMOUSLY. MAKE SURE
THAT YOU DO NOT INDICATE YOUR NAME ANYWHERE IN YOUR
SUBMISSION.
The work should be uploaded as a single PDF file. You are advised to cut and paste your
answers for each question into a single Word document and then save it as a PDF which you
can then upload. This takes time so make sure that you leave enough time to do this, and that
you have tried out the upload system in advance so that you do not run into unexpected
difficulties at the last minute. I strongly recommend that you upload versions of your
work as you go along, so that if there are last-minute difficulties you will at least have
submitted an early version of your work. You can submit as many versions as you like: only
the last version submitted before the deadline will be marked.
For all of these questions you will receive some marks for incomplete answers. If you
cannot get your programme to work, you should submit it anyway. You will get credit
for your attempt, and you will not lose many marks if an obscure bug was stopping an
essentially correct programme from working!
In these questions “Banner number” means your nine-digit Banner number, which you can
find on your Greenwich Gateway card. This begins with 000 and for our purposes we do not
include the tenth extra check digit after the hyphen.
In all questions you will lose marks if your output is not clear. You are expected to use
the facilities of Matlab and Excel to ensure that graphs and charts are well presented
and clearly labelled. Your programmes should give the user clear instructions and be
designed to be as easy to use as possible, and marks will be deducted for weaknesses in
this regard. Matlab programmes should have plentiful comments to explain how the
programme works and you will lose marks if your comments are inadequate.
There are EIGHT questions. You should attempt every question.
1. Use Matlab to carry out the following calculations. Submit a copy of the Matlab
command window. Do not worry if you make mistakes and generate error messages
as long as the correct calculations and answers are there and clearly indicated.
Find the value of the function ??(??) = ????
?? – 9???? + 12?? + (??????)
?? v?? when x=0.2 and
when x=0.7653.
Evaluate ???????? (????
?? )
Change the following two complex numbers into polar form: 12 + 5i and 4 – 3i
Display exactly the first six digits (only) of p/2.
[5 marks]
2. Create a script file that you will be able to use at the end of this academic year to
calculate your final mark for MATH1103 MaTT. It should present you with a
message alerting you if you have either done particularly well (a mark of 70 or higher,
which is on course for First Class honours) or particularly badly (a mark below 40,
which is a fail).
You should include all five items of assessment with the relevant percentage
weighting which you can find in the course specification or on the Moodle page.
A basic programme which performs the calculation successfully will receive 8 marks,
Extra marks will be awarded for script files that are designed to be easy to use – so
make good use of Matlab’s input and output commands – and/or which provide other
useful information to the user, and for appropriate testing.
Submit a printscreen of the output after running your programme for your test cases as
well as a copy of the script file itself. Obviously you do not yet know all your marks
so test it with a suitable range of possible marks.
[15 marks]
3. Go to http://www.metoffice.gov.uk/climate/uk/stationdata/
If your family name begins with A select Armagh
If your family name begins with B-C select Bradford
If your family name begins with D-G select Durham
If your family name begins with H-J select Heathrow
If your family name begins with K-L select Lerwick
If your family name begins with M-O select Oxford
If your family name begins with P-S select Sheffield
If your family name begins with T-Z select Yeovilton
Save these data as a .txt file and open it in Excel, following the prompts.
Select data from a year of your choice ending with the same number as the 9th digit of
your Banner number. In this year, start from April if your birthday is in the first half
of the year (January – June inclusive) and start from September if your birthday is in
the second half of the year (July to December inclusive). Take data for the next 12
months from this starting point.
Copy these data on to a separate Excel sheet and save it.
Import these data into Matlab, using the method given in the lecture notes, and
create a graph showing the maximum and minimum temperatures for your data. Use
‘legend’ to produce a suitable key.
Submit a copy of the command window showing the commands for importing the
data and the commands for producing the graph, and also submit a copy of the actual
graph.
Marks will be awarded for presentation as well as for importing the data correctly
from Excel so aim for a nicely-presented graph that is well-labelled.
[10 marks]
4. If the 8th digit of your Banner number is odd, do part A: if it is even, do part B.
In each case submit your script along with a copy of the command window showing
your results.
A) Use a Monte Carlo approach to calculate the area of the region bounded by the
lines x=0, y=0 , x=1 and y=x3. In other words you have to generate random
pairs of points (x,y) with 0 < x,y < 1 and see what proportion satisfy y < x3, to
get an estimate of the proportion of the area of the region you want to that of
the unit square. (You can calculate the exact answer by calculus!)
Run your programme for 10, 100, 1,000, 10,000, 100,000 and 1,000,000 trials.
Which gives the most accurate result? Is that what you expected?
B) Use a Monte Carlo approach to calculate the area of the part of the parabola
bounded by the lines x=0, y=1, x=1 and y = sin(px/2) In other words you
have to generate random pairs of points (x,y) with 0 < x,y < 1 and see what
proportion satisfy y < sin(px/2), to get an estimate of the proportion of the area
of the region you want to that of the unit square. (You can calculate the exact
answer by calculus!)
Run your programme for 10, 100, 1,000, 10,000, 100,000 and 1,000,000 trials.
Which gives the most accurate result? Is that what you expected?
(10 marks)
5. For this question you should do:
Part A if the ninth digit of your Banner number is 0, 1, 2 or 3;
Part B if it is 4, 5 or 6;
Part C if it is 7, 8 or 9.
For whichever part you have to attempt, you must submit both your m-file and the
output it produces when run. If you have attempted a programme but cannot get it to
run, submit what you have done so far and say what happens when you run it.
Explain what you have tried to do and what you want to do. You will gain marks for
the understanding you show even if you cannot get your programme to run. To
achieve the highest marks your programme should be written elegantly, have been
tested for a suitable range of examples, and produce clear understandable and concise
output.
A) Write a programme which asks the user for a number between 1 and 1000 and
calculates the sum of its digits – for example, if the number is 987 then the
sum of its digits would be 24.
B) Write a programme which asks the user for a number between 1 and 1000 and
works out whether it is divisible by each of the numbers 2, 3, 5 and 7. For
example, if the number is 110 the programme should say that it is divisible by
2 and 5: if it is 101 the programme should say that it is not divisible by any of
these numbers.
C) Write a programme which asks the user for a number between 1 and 500 and
works out whether the last digit of the number is the same as the last digit of
its cube. For example, if the number is 10, then 10 and 103 = 1000 share the
same last digit, while if the number is 17, then 17 and 173 = 4913 have
different last digits.
[10 marks]
6. Use Excel to plot the graph of f(x)=2sin(2x/3)-3cos(3x).
Add a title, label the axes and label the function. Make the axes thicker than the other
grid lines. Choose a suitable range of x and y values that shows at least two roots of
the function.
Find the value of the first positive root correct to 5 decimal places using the Newton-
Raphson method on Excel. Use stopping criteria |xn- xn+1|<0.0001
Also find the root using the Bisection Method, with the same stopping criterion.
Submit the graph and two versions of your Excel worksheets containing the Newton-
Raphson method and stating the answer, one showing the numbers and the other
showing the formulae you have used.
For this question you may start from the Excel worksheets given in the lecture notes.
For full marks you should use IF formulae to indicate when the stopping criteria have
been met.
[15 marks]
7. Create an Excel spreadsheet to investigate the behaviour of the logistic equation
xn+1 = k xn (1 – xn)
for various values of k. In each case start with a suitable initial value for x0: choosing
x0 = 0.5 is good in most cases. Use an appropriate Excel formula to calculate
successive values for xi and see what happens. Does it converge to a single value? If
not, what does happen?
Try values of k such as 2, 3, 3.2, 3.5 and 3.6.
Comment on what happens, and explore the values of k for which different things
happen. You may use any tools you wish in Excel, and you may also research the
topic online and/or use Matlab for further exploration. Submit an account of your
investigation, including Excel output, and also include copies of the worksheets
showing the formulae you have used.
An investigation which shows what happens for the listed values (with a welldesigned
worksheet) will obtain 15 marks. To obtain higher marks your investigation
must go beyond that. If you are presenting the results of research, make sure that you
tell me your sources.
[30 marks]
8. Carry out the following matrix calculations using the appropriate matrix commands in
Excel where the matrices A, B, C and D are as below:
A=?? 5 8
-2 6
?? B=??7 -8
4 12?? C=??
5 1 -6
2 -2 1
-3 5 2
?? D=??
-3 1 5
2 0 7
1 -1 1
??
• AB
• Determinant of (AB)
• (CD)-1 (that is, the inverse of CD)
• D-1C-1
• D3
Hand in two printouts of the Excel sheet, one with the answers and one showing the
formulae you have typed in.
[5 marks]
Assessment criteria for the assignment: Marks will be awarded as indicated for each
question. Some marks will be awarded for partial or incomplete solutions so, if you cannot
complete a question, you should still hand in whatever you have done.