## 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]

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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]