Analytical Methods for Computing

Analytical Methods for Computing
Luckily there are no secrets to this assignment. All that is required is for the questions to be answered. The first page and a half is a header and can be ignored. A lot of the questions is easier to be answered by hand, so that is fine. It can all be done by hand if that’s necessary.
MATH1111 Page 1 of 5 YDF
MATH1111
(2016/17)
Analytical Methods for
Computing
Faculty Header ID:
300367
Banner Header ID for
handin: 234482
Contribution: 50% of
course
Course Leader:
Dr Yvonne Fryer
Assignment Deadline Date:
Friday 24/03/2017
This coursework will be marked anonymously
YOU MUST NOT PUT ANY INDICATION OF YOUR IDENTITY IN YOUR SUBMISSION
This coursework should take an average student who is up-to-date with tutorial work
approximately 25 hours
Feedback and grades are normally made available within 15 working days of the coursework
deadline
Learning Outcomes:
A. Use functions in the context of computing.
B. Design and use simple algorithms.
C. Use vectors and matrices in a variety of applications.
D. Understand small network graphs and apply them to a variety of problems.
E. Understand some basic concepts of differential and integral calculus and apply them in the
context of computing.
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.
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.
By handing in your coursework to the Faculty Reception Desk you are confirming that it has
not, in whole or part, been presented elsewhere for assessment.
In addition, you are confirming that
? All material which has been copied has been clearly identified, for example, by being
placed inside quotation marks and a full reference to the source has been provided
? Any material which has been referred to or adapted has been clearly identified and a full
reference to the source has been provided
? Any work not in quotation marks is in your own words
? You have not shared your work with any other student, unless this was a group
assignment in which case it has only been shared with members of the group when
necessary
MATH1111 Page 2 of 5 YDF
? You have not taken work from any other student
? You have not paid anyone to do your work or employed the services of an essay or code
writing agency
Coursework Submission Requirements
? A paper copy of your work for this coursework must be submitted to the Faculty Reception
Desk before the office closes on the Deadline Date of Friday 24/03/2017.
? You must submit your coursework with a bar-coded coursework header that YOU MUST
PRINT YOURSELF. Just follow the instructions and print your header sheet for this
assessment item and submit it with your coursework.
? Hand your work with the Header Sheet to the Faculty Reception Desk.
? There is no facility to upload an electronic copy of your work for this coursework.
? Remember to keep your coursework receipt.
? 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
Detailed Specification
The coursework is to be completed individually
Deliverables
Students need to produce a piece of written work that includes all the working out and answers
for the following questions.
Grading Criteria
Grades can be easily worked out by looking at the marks allocated for each part of the
coursework. An excellent coursework is one that is complete, has few errors and includes working
and therefore will receive a mark in excess of 70%. A coursework that is just a pass at 40% will
have multiple errors, working missing and possibly even questions missing. To be certain of getting
the best mark include all working and attempt all questions.
Assessment Criteria
The marks allocated per section of the coursework are identified in brackets by the side of a
question.
MATH1111 Page 3 of 5 YDF
Answer all questions, working needs to be shown to get full marks.
? At the top of your page write down your banner
student id., starting with three zeros. i.e., 0 0 0 9 0 2 4 8 2
? Ignore the first three zeros, replace any remaining
0’s by 1’s. i.e., 9 1 2 4 8 2
? Order your id. from smallest to largest. i.e., 1 2 2 4 8 9
? Label the values A-F as shown ?? ?? ?? ?? ?? ??
? Write down the values of ?? to ??, for your student id. on your coursework submission.
? Use the values for ?? to ?? in the following questions where applicable as constants.
1: Functions
a) The following functions are defined for all real numbers. Sketch the graphs and find
the range of each function.
i. ??(??) = ???? + 3 ii. ??(??) = ??
2 – ?? iii. ??(??) = sin(????) + 1
[9 marks]
b) Find the inverse of the following functions
i. h: ?? ? ??, where ?? = {????h??????, ????????????, ????????????, ??????????},
?? = {214, 580, 786, 1488} and
h(????h??????) = 1488, h(????????????) = 580, h(????????????) = 786, h(??????????) = 214
ii. ??: ?? ? ??, where ??(??) = ???? – ??
[6 marks]
c) Explain why it is not possible to find an inverse function of ??(??) = ????
2 + 1
[2 marks]
d) Describe function ??(??) = ????
2 + ???? + ?? using the: into/onto and one-one/manyone/one-many
notation.
[3 marks]
MATH1111 Page 4 of 5 YDF
2: Matrices
a) Write down the polygon ?? such that ?? = (
0 ??
0 3
?? + 2 ?? + 1 ??
0 -3 -3
) and each
column of ?? represents a (??, ??) vertex of the polygon.
Draw its position on a graph.
Write down a matrix ?? that will reflect the polygon along the line ?? = ??.
Use the transformation on ?? and draw the transformed Polygon on the graph.
[8 marks]
b) Where possible solve the following system of equations, using the inverse of a
matrix:
???? + ???? = ?? – 2?? – 3??
???? – ???? + 8 = ?? – 2??
Plot the two equations on a graph and confirm the results. If it is not possible to
solve the system of equations you need to explain why not.
[12 marks]
3: Graph theory
a) The following distance table shows the distance in kilometres between some cities
in the USA.
Distance
(Km)
Boston Chicago Los
Angeles
Miami New York San
Francisco
Boston 1589 4891 2474 342 5067
Chicago 3366 2184 1352 3493
Los Angeles 4373 4539 667
Miami 2133 4990
New York 4826
i. Draw this information as a graph.
ii. When visiting the USA I would like to visit all of these places. Would Kruskal’s
algorithm help determine a route through the cities to minimise the distance
travelled? Explain your answer – a simple yes/no will not get the marks.
[8 marks]
b) For the graph shown here determine
the adjacency matrix and determine a
matrix that contains information,
starting point and end point, on all
walks of length 3.
[12 marks]
u v
w
x
y z
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4: Algorithms
a) Investigate the following algorithm:
Algorithm Unknown
READ ??, ??,??h??????, ??
??_???????? ? cos(??h??????) × ?? – sin(??h??????) × ??
??_???????? ? sin(??h??????) × ?? + cos(??h??????) × ??
?? ??? × ??_????????
?? ??? × ??_????????
DISPLAY ??, ??
END Unknown
where ??, ??,??h??????, ?? are real numbers. Explain what the algorithm does and what the
output values are.
Modify the algorithm to allow ??, ?? to be arrays. What would the purpose of the
algorithm be then?
[20 marks]
5: Differentiation/Integration/Calculus
a) Differentiate the following functions
i. ?? = ????
2 – ???? + ????
??
ii. ?? = ??
??
sin(????)
[6 marks]
b) Given
??(??) = ???? – ??
2
i. Plot the function ??(??) over the interval [0, ??]
ii. Determine the value of the following definite integral ? ??(??)
??
0
????. Show
what this means on the graph in 5.b)i.
iii. Differentiate ??(??) and determine the value of ????
????
when ?? = 0. What does ????
????
represent?
[14 marks]

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