(a) Use the Recurrent relation for generating the Chebyshev polynomials which is given by To(x) = 1, T1(x) = x, Tn+1(x) = 2xTn(x) –Tn-1(x) to  approximate the function   with a Maclaurin’s polynomial of order 5. Hence use the Chebyshev polynomials and method of economization to obtain a lesser degree polynomial of approximation, of order 3.

(b) Determine the upper error bound, due to the economization.

(c) Compare and contrast with results obtained using Legendre’s orthogonal polynomials.

United Business Institutes Bachelor of Arts in Business (Honours) Programme
COR140 Maths and Statistics Final Exam – Version A
This is the final exam for COR140 Mathematics and Statistics with a pass mark of 50%. You have 2
hours to finish the exam and you must answer ALL the questions to get full marks.
You may NOT use a graphical scientific calculator nor one capable of algebraic manipulation. The
use of mobile computers and/or communication devices, of any type, capable of storing information
or retrieving information from any external source, including, but not limited to, the World Wide Web
(WWW), is expressly forbidden by the Institute’s Examination Regulations. Anyone breaking this
rule will have their exam script declared null and void.
When answering the questions show your working out so that you can be credited for incomplete
answers or even answers that are mostly correct.
Student Name
UBI Campus
This exam covers all the Learning Objectives for the course as stated in the Module Narrative.
Question Points Score
1 9
2 5
3 9
4 10
5 13
6 14
Total: 60
COR 140 Mathematics and Statistics I Final Exam
1. (a) Find: (1 mark)
log4(1024)
(b) Find: (2 marks)
ln(e
4t
)
(c) Calculate the point of intersection of the two straight lines, y = 3x + 6 and y = 2x + 9 (3 marks)
(d) Evaluate the definite integral below, giving your answer as a top-heavy fraction with denom- (3 marks)
inator 3.
Z 5
3
(x
2
) dx
COR140 – Mathematics and Statistics I – Final Version A Page 2 of 8
COR 140 Mathematics and Statistics I Final Exam
2. Given the function: x
2 − 3x + 2
(a) Find the roots of the equation by factorisation (show your working out). (3 marks)
(b) Find the coordinates of the minimum of the function. (2 marks)
COR140 – Mathematics and Statistics I – Final Version A Page 3 of 8
COR 140 Mathematics and Statistics I Final Exam
3. (a) What is the value of an investment of e2500 at 3.5% compounded annually for:
i. 2 years? (1 mark)
ii. 6 years? (1 mark)
(b) Your grandfather has placed e5500 into an investment fund paying 6% a year. How long (3 marks)
will it take for the fund to reach e9000? Give your answer in years and months rounded up
to the month above.
(c) Calculate the daily interest rate needed to increase your grandfather’s investment from (4 marks)
e5500 to e9000 over 5 years. Assume that a financial year comprises 360 days. Give your
answer to 4 decimal places.
COR140 – Mathematics and Statistics I – Final Version A Page 4 of 8
COR 140 Mathematics and Statistics I Final Exam
4. You have made an investment based on the terms and conditions given by your financial institution. Your mathematical education at UBI means that you are able to derive a simple exponential
function to model the investment conditions. The function that you derive for an investment of
e3000 is:
g(t) = 2.5e
1.05t + 3000 (1)
Using this model:
(a) What is the value of your investment after 3 years? (2 marks)
(b) Find the first derivative of the function, g(t). (3 marks)
(c) How fast is the investment growing per year at:
i. 4 years? (2 marks)
ii. 5 years? (1 mark)
(d) Are these good investment terms? Explain your answer. (1 mark)
(e) Are these realistic investment terms? Explain your answer. (1 mark)
COR140 – Mathematics and Statistics I – Final Version A Page 5 of 8
COR 140 Mathematics and Statistics I Final Exam
5. The Harrison Watch Company are planning an expensive, limited edition “Longitude” timepiece.
The price demand and average cost functions (in thousands of euros per timepiece) modelled by
the chief economist in the company’s marketing department are:
P + Q = 25
and
AC = 3 +
21
Q
where Q is the number of timepieces sold and P is the price.
(a) Derive an expression for the profit function in this model. (3 marks)
(b) What is the order of the profit function polynomial? (1 mark)
(c) Evaluate the discriminant of the profit function. (2 marks)
(d) Find the roots of the profit function. (2 marks)
(e) What is the lowest number of timepieces that the company needs to sell in order to break (2 marks)
even?
(f) How many timepieces must the company sell to maximize its profits? (2 marks)
COR140 – Mathematics and Statistics I – Final Version A Page 6 of 8
COR 140 Mathematics and Statistics I Final Exam
(g) What is the maximum profit that the company can generate according to the chief econom- (1 mark)
ist’s model?
COR140 – Mathematics and Statistics I – Final Version A Page 7 of 8
COR 140 Mathematics and Statistics I Final Exam
6. Mr Fisher and Mr Spearman play Scrabble1
at least three times a week. In the game, a bonus of
50 points is awarded if one can use all seven letter tiles in one turn. They record the number of
times that either of them uses all 7 tiles during a turn. The weekly figures for each of 18 weeks
are shown in the table below.
Week 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
7-tile Turns 7 9 7 10 8 12 19 8 7 9 7 6 4 7 6 10 11 9
(a) Compute the minimum, first quartile, median, third quartile, maximum and range of the (3 marks)
weekly scores.
(b) On squared (graph) paper, draw a labelled box plot that includes all of the scores. (4 marks)
(c) Compute the IQR and state whether any of the data points are outliers. Explain how you (2 marks)
identify an outlier.
(d) Draw a second box plot, on the same page as the first, that excludes the outlier. (3 marks)
(e) Calculate the population standard deviation for the scores. (2 marks)
1Scrabble is a game where players try to create words on a 15-by-15 board using from 1 to 7 letters tiles from their letter
rack.
COR140 – Mathematics and Statistics I – Final Version A Page 8 of 8

MATH 270 TEST 4 REVIEW
1. Let A = P DP −1 and compute A4 where P =
”
5 7
2 3 #
and D =
”
2 0
0 1 #
.
2. Diagonalize the following matrix where the eigenvalues are λ = 5, 1.



2 2 −1
1 3 −1
−1 −2 2



3. Let T : P2 → P3 be the transformation that maps a polynomial p(t) into the polynomial (t + 5)p(t). Find the matrix for T relative to the bases {1, t, t2} and {1, t, t2
, t3}.
4. Let the following matrix act on C
2
. Find the eigenvalues and a basis for each eigenspace
in C
2
. ”
1 5
−2 3 #
5. Find an invertible matrix P and a matrix C of the form ”
a −b
b a #
3
the given matrix has the form A = P CP −1
. Use the information from problem 4.
”
1 5
−2 3 #
6. Find the distance between x =
”
10
−3
#
and y =
”
−1
−5
#
.
7. Let u =



2
−5
−1



and v =



−7
−4
6


 . Compute ku + vk
2
.
8. Compute the orthogonal projection of ”
1
7
#
onto the line through ”
−4
2
#
and the origin.
9. Let y =
”
2
3
#
and u =
”
4
−7
#
. Write y as the sum of two orthogonal vectors, one in Span{u}
and one orthogonal to u.
10. Let y =
”
3
1
#
and u =
”
8
6
#
. Compute the distance from y to the line through u and the
origin.
11. Determine whether the set of vectors are orthonormal. If the set of vectors are only
orthogonal, normalize the vectors to produce an orthonormal set.
(Rationalize your denominator, if necessary).








1
3
1
3
1
3








,








−
1
2
0
1
2








12. Find the orthogonal projection of y onto the Span{u1, u2}.
y =



−1
2
6


 , u1 =



3
−1
2


 , u2 =



1
−1
−2



13. Let W be the subspace spanned by the u’s and write y as the sum of a vector in W
and a vector orthogonal to W.
y =



1
3
5


 , u1 =



1
3
−2


 , u2 =



5
1
4



14. Find an orthogonal basis for the column space of the following matrix.





3 −5 1
1 1 1
−1 5 −2
3 −7 8





15. Let R
2 have the inner product given by hx, yi = 4x1y1 + 5x2y2 3 x = (1, 1) and
y = (5, −1). Compute kxk , kyk and |hx, yi|2
.
16. Let P2 have the inner product given by evaluation at −1, 0 and 1. Compute hp, qi
where p(t) = 4 + t, q(t) = 5 − 4t
2
.
17. Based on problem 16, compute kpk and kqk.
18. For f, g ∈ C[0, 1], let hf, gi =
Z 1
0
f(x)g(x)dx. Compute h1 − 3t
2
, t − t
3
i.
19. Based on problem 18, compute kfk. (Rationalize your denominator, if necessary).
20. Find the third-order Fourier approximation to f(t) = 2π − t.

MATH2033
Introduction to Scientific Computation
Coursework 2
This coursework is due by 23:59 on Friday the 5th of June 2020
The folder containing this file also contains the files interpolant.m, population.m, differentiation.m,
integration.m, forwardEuler.m and errorEuler.m. To do this coursework, you should add to
these files but should not rename them and should not modify the lines that are already in them.
Via Moodle, you should submit a single compressed folder (in 7z, gz, tar, or zip format) whose name
is your student number and which contains your completed version of these 6 files. Each question is
worth 2.5% of your final mark in this module. However, a penalty may be applied to your mark in
this coursework for not following these instructions and the standard penalties for late submissions
will be applied.
This coursework should be your own individual work. However, it is acceptable for parts of
your codes to have been copied from the MATLAB files that are on the Introduction to Scientific
Computation (MATH2033 UNNC) (FCH1 19-20) Moodle page. It is also acceptable for you to ask
the MATH2033 lecturer questions about the coursework during class. If he decides to answer your
question then the answer will be given to all students in attendance. The lecturer will not answer
any questions about the coursework that are asked outside of class.
1. Add to the file interpolant.m to write an interpolant function that does what is described
in this paragraph. The arguments to the interpolant function are vectors a and b with at
least n+1 entries which are such that the first n+1 entries of a are distinct, a positive integer
n and a vector x. The interpolant function should return the vector p which has the same
number of entries as x and is such that, for all positive integers i which are less than or equal
to the number of entries in x, p(i) is the value at x(i) of the polynomial of degree at most
n which passes through the points (a(j),b(j)) for all positive integers j which are less than
or equal to n+1. The interpolant function should not display anything to the command
window if the line in which it is used ends with a semicolon.
2. The population of mainland China, according to the censuses conducted, in the years 1953,
1964, 1982, 1990, 2000 and 2010 is given in the table below.
year population of mainland China
1953 582603417
1964 694581759
1982 1008175288
1990 1133682501
2000 1265830000
2010 1339724852
Add to the file population.m to write a population script that uses the interpolant
function to plot:
• the linear polynomial that passes through the data points for the years 1953 and 1964;
• the quadratic polynomial that passes through the data points for the years 1953, 1964
and 1982;
• the cubic polynomial that passes through the data points for the years 1953, 1964, 1982,
1990;
• the quartic polynomial that passes through the data points for the years 1953, 1964,
1982, 1990 and 2000;
• the quintic polynomial that passes through the data points for the years 1953, 1964,
1982, 1990, 2000 and 2010.
on the same figure for the years from 1950 to 2020. You should use a different colour for each
polynomial and should not use black to plot the polynomials. On top of these polynomials
plot the data given in the table as six individual points that are circles whose boundary and
interior are black. Label the axes. Put a legend on the figure which does not cover up anything
that you have plotted. Your legend should identify each polynomial as “linear interpolant”,
“quadratic interpolant”, “cubic interpolant”, “quartic interpolant” or “quintic interpolant”.
The population script should not display anything to the command window. You should not
submit your plot.
3. Add to the file differentiation.m to write a differentiation function that does what is
described in this paragraph. The arguments to the differentiation function are a function
handle f of a function that implements an appropriate function f, a real number x, a positive
real number h and a number n which can be −1, 0 or 1. Using no more of the values of f than
f(x−h), f(x) and f(x+h), the differentiation function should return the approximation
a to f
′
(x) that is:
• the backward difference approximation to f
′
(x) if n= −1;
• the central difference approximation to f
′
(x) if n= 0;
• the forward difference approximation to f
′
(x) if n= 1.
The differentiation function should not display anything to the command window if the
line in which it is used ends with a semicolon.
4. Add to the file integration.m to write a integration function that does what is described
in this paragraph. The arguments to the integration function are a function handle f of a
function that implements an appropriate function f, real numbers a and b which are such that
a<b, a positive integer n and a number m which can be 0, 1 or 2. The integration function
should return the approximation A to Z
b
a
f(x) dx obtained using:
• the composite midpoint rule with n strips of equal width if m= 0;
• the composite trapezoidal rule with n strips of equal width if m= 1;
• the composite Simpson’s rule with n strips of equal width if m= 2.
The integration function should not evaluate f any more times than is necessary. The
integration function should not display anything to the command window if the line in
which it is used ends with a semicolon.
5. Let y be the solution to the initial value problem
y
′
(t) = f(t, y(t)), a ≤ t ≤ b,
y(a) = c.
Add to the file forwardEuler.m to write a forwardEuler function that does what is described
in this paragraph. The arguments to the forwardEuler function are a function handle f of a
function that implements an appropriate function f, real numbers a and b which are such that
a<b, a real number c and a positive integer N. The forwardEuler function should return the
approximation yN to y(b) obtained using the forward Euler method with a stepsize of b − a
N
.
The forwardEuler function should not display anything to the command window if the line
in which it is used ends with a semicolon.
6. Let y be the solution to the initial value problem
y
′
(t) = f(t, y(t)), a ≤ t ≤ b,
y(a) = c.
Add to the file errorEuler.m to write a errorEuler function that does what is described
in this paragraph. The arguments to the errorEuler function are a function handle f of a
function that implements an appropriate function f, a function handle y of a function that
implements y, real numbers a and b which are such that a<b, a real number c and a positive
integer N. The errorEuler function should use the forwardEuler function to plot a loglog
plot of the absolute value of the error in the approximation to y(b) obtained using the forward
Euler method with a stepsize of h =
b − a
2
0
,
b − a
2
1
, . . . ,
b − a
2
N
on an appropriately labelled
figure. The errorEuler function should not return anything. The errorEuler function
should not display anything to the command window. You should not submit your plot.