tw36 university of bolton school of engineering b… · ... automotive performance engineering...

TW36

UNIVERSITY OF BOLTON

SCHOOL OF ENGINEERING

B.ENG (HONS) AUTOMOTIVE PERFORMANCE ENGINEERING

EXAMINATION SEMESTER 2 - 2016/2017

ENGINEERING MATHEMATICS

MODULE NO: MSP4007

Date: Wednesday 17 May 2017 Time: 2.00 – 4.00

INSTRUCTIONS TO CANDIDATES: There are FOUR questions.

Answer ALL questions.

All questions carry equal marks.

Marks for parts of questions are shown in brackets.

Electronic calculators may be used provided that data and program storage memory is cleared prior to the examination.

CANDIDATES REQUIRE: Formula Booklet and MATLAB

Booklet Provided.

PAST EXAMIN

ATION P

APER

of 5 School of Engineering B.Eng.(Hons) Automotive Performance Engineering Examination Semester 2 - 2016/2017 Engineering Mathematics Module No. MSP4007

Question 1

3

3 5

Differentiation - Differentiate the following:

a)

( ) (1 mark)

1( ) ( ) (1 mark)

( ) ( )

i f x x

ii f xx

iii f x x

9

3 2

22

(1 mark)

b)

( ) ( ) 3 (2 marks)

6( ) ( ) (2 marks)

c)

3( ) ( ) 5 7 8 (2 marks)

(

i f x x

ii f xx

i f x x xx

ii

24

2 2

2 6

2

2

1) ( ) 2 (2 marks)

d)

1( ) (3 marks)

( 4 5)

e)

( ) ( 6)(4 5) (3 marks)

f)

2( )

2

f x x xx

f xx x

f x x x

xf x

x

2

(3 marks)

g)

( ) exp( 4) (2 marks)

h)

( ) sin(2 )cos(4 ) (3 marks)

f x x

f x x x

Total 25 marks Please turn the page

PAST EXAMIN

ATION P

APER


Question 2

3

3 5

Integration - Integrate the following:

a)

( ) ( ) (1 mark)

1( ) ( ) (1 mark)

( ) ( )

i f x x

ii f xx

iii f x x

3 2

4

3

(1 mark)

b)

( ) ( ) 7 -3 2 24 (2 marks)

2( ) ( ) 3 (2 marks)

c)

( ) ( ) (3 8) (2 marks)

( ) ( ) 6 7

i f x x x x

ii f x xx

i f x x

ii f x x

2

2

(2 marks)

d)

( ) 2 sin 6 (3 marks)

e)

( ) exp(4 5) cos(4 5) (3 marks)

f)

2 1( ) (3 marks)

3 2

f x x x

f x x x

xf x

x x

7

g)

( ) (3 5)(2 1) (5 marks)f x x x

(Total marks:25) Please turn the page

PAST EXAMIN

ATION P

APER


Question 3 Section 1: Partial Fractions

(a) Find the partial fractions of:

[8 marks]

(b) Check your answer by adding the partial fractions together again. [2 marks]

Section 2: Cramer’s rule Solve the following system of three linear equations using Cramer’s rule:

13 2 7

2 0.5

[15 marks]

[Total Marks: 25] Question 4 i) If a = [5 2 0 3 -4], b = [0 1 4 8 2] and c = 2, perform the following operations using Matlab and provide the results. a) Add vectors a and b. b) Raise each element of vector a to c. c) Create a new vector d consisting of the fourth, third and second elements of vector b in this particular order. Hint: vector (first element: step: last element) [10 marks] ii) Develop a Matlab script using the function ones in order to form a

(5 rows) x (2 columns) matrix, in which each element equals to 9 i.e.

9 99 99 99 99 9

.

[5 marks]

Question 4 continued over the page

PAST EXAMIN

ATION P

APER

of 5 School of Engineering B.Eng.(Hons) Automotive Performance Engineering Examination Semester 2 - 2016/2017 Engineering Mathematics Module No. MSP4007 Question 4 continued iii) Sketch using Matlab a cosine wave starting from point zero ending at point 12.57 (4π) consisting of 200 points. In your plot, include the x-label: ‘Independent variable x’, the y-label: ‘Dependent variable y’ and the title: 'Cosine curve'.

[10 marks] Note that for all the questions above you need to include in your answer both the Matlab code as well as the corresponding output.

[Total Marks: 25]

END OF QUESTIONS

PAST EXAMIN

ATION P

APER

Matlab CourseMatlab Course

Ioannis Paraskevas (PhD, CEng)

[email protected]

Array OperationsArray Operations

� Definition - Simple Arrays

� Array Addressing

� Array Construction

� Array Mathematics

� Arrays of Ones or Zeros

Definition Definition -- Simple ArraysSimple Arrays

Operations involving scalars are the basis of mathematics. At the same time, when we wish to perform the same operation on more than one number at a time, repeated scalar operations are time-consuming and cumbersome.To solve this problem, MATLAB defines operations on data arrays.

ExampleExample

� Compute the values of the sine function over one-half of its period, namely y=sin(x), x belongs [0,pi].

Forx = 0, .1pi , .2pi , .3pi , .4pi , .5pi , .6pi , .7pi, .8pi, .9pi,

.pi

y = 0, .31, .59, .81, .95, 1.0, .95, .81, .59, .31, 0

In MATLAB:

>> x=[0 .1*pi .2*pi .3*pi .4*pi .5*pi .6*pi .7*pi .8*pi .9*pi pi] (Enter)

x =

0 0.3142 0.6283 0.9425 1.2566 1.5708 1.8850 2.1991 2.5133 2.8274 3.1416

>> y=sin(x) (Enter)

y =

0 0.3090 0.5878 0.8090 0.9511 1.0000 0.9511 0.8090 0.5878 0.3090 0.0000

Array AddressingArray Addressing

� In MATLAB, individual array elements are accessed using subscripts, e.g.,x(1) is the first element in x, x(2) is the first

element in x etc. For example:>> x(3) % The third element of x

ans = 0.6283

>> y(5) % The fifth element of y

ans = 0.9511

Array AddressingArray Addressing

� To access a block of elements at one time, MATLAB provides colon notation. For example:

>> x(1:5) % start with one and count up to 5

ans = 0 0.3142 0.6283 0.9425 1.2566

>> x(7:end) % starts with the seventh element and continues to the last element

ans = 1.8850 2.1991 2.5133 2.8274 3.1416

>> y(3:-1:1) % start with 3, count down by 1, and stop at 1

ans = 0.5878 0.3090 0

>> x(2:2:7) % start with 2,count up by 2, and stop when you get to 7

ans = 0.3142 0.9425 1.5708

>> y([8 2 9 1]) % extracts the elements of the array y in the order we wish

ans = 0.8090 0.3090 0.5878 0

Array ConstructionArray Construction

The basic array construction features of MATLAB are summarized as follows:Basic Array Construction

x = [2 2*pi sqrt(2) 2-3j] create row vector x containing elements specifiedx = first:last create row vector x starting with first, counting by one, ending at

or before lastx = first:increment:last create row vector x starting with first, counting by

increment, ending at or before lastx = linspace(first,last,n) create row vector x starting with first, ending at last,

having n elementsx = logspace(first,last,n) create logarithmically spaced row vector x starting

with 10^(first), ending at 10^(last), having n elements

ExamplesExamples

>> x=(0:0.1:1)*pi % create row vector x starting with first, counting by increment, ending at or before last

x = 0 0.3142 0.6283 0.9425 1.2566 1.5708 1.8850 2.1991 2.5133 2.8274 3.1416

>> x=linspace(0,pi,11) % create row vector x starting with first, ending at last, having n elements

x = 0 0.3142 0.6283 0.9425 1.2566 1.5708 1.8850 2.1991 2.5133 2.8274 3.1416

>> x=logspace(0,2,11) % create row vector x starting with first, ending at last, having n elements

x = 1.0000 1.5849 2.5119 3.9811 6.3096 10.0000 15.8489 25.1189 39.8107 63.0957 100.0000

Array MathematicsArray Mathematics

The basic array operations are summarized as follows:Element-by-Element Array Mathematics

Illustrative data: a = [a1 a2… an], b = [b1 b2… bn], c = <a scalar>� Scalar addition: a + c = [a1+ c a2+ c… a2+ c ]� Scalar multiplication: a * c = [a1* c a2* c … an* c ]� Array addition: a + b = [a1+ b1 a2+ b2… an+ bn ]� Array multiplication: a .* b = [a1* b1 a2* b2 … an* bn ]� Array division: a ./ b = [a1 / b1 a2 / b2 … an / bn ]� Array powers: a.^c = [a1 ̂ c a2 ̂ c … an ̂ c]

c.^a = [c^ a1 c^ a2 … c^ an]a.^b = [a1 ̂ b1 a2 ̂ b2 … an ̂ bn ]

Arrays of Ones or ZerosArrays of Ones or Zeros

� Because of their general utility, MATLAB provides functions for creating arrays containing either all ones or all zeros

Examples

� >> ones(3) % creates a 3x3 matrix consisted of 1s

ans =

1 1 1

1 1 1

1 1 1

� >> zeros(2,5) % creates a 2x5 matrix consisted of 0s

ans =

0 0 0 0 0

0 0 0 0 0

Relational and Logical Relational and Logical OperationsOperations

Relational operatorsRelational Operator Description

� < less than

� <= less than or equal to

� > greater than

� >= greater than or equal to

� = equal to

� ~= not equal to

ExamplesExamples

>> A=1:9

A =

1 2 3 4 5 6 7 8 9

>> B=9-A

B =

8 7 6 5 4 3 2 1 0

>> tf=A>4 % finds elements of A that are greater than 4.

tf =

0 0 0 0 1 1 1 1 1

>> tf=B-(A>2) % finds where A>2 and subtracts the resulting vector from B

tf =

8 7 5 4 3 2 1 0 -1

Relational and Logical Relational and Logical OperationsOperations

Logical operatorsLogical Operator Description� & AND� | OR� ~ NOT

Example>> A=1:9A =1 2 3 4 5 6 7 8 9>>tf=(A>2)&(A<6) % returns ones where A is greater than 2 AND less than 6tf =0 0 1 1 1 0 0 0 0

Other Relational and Logical Other Relational and Logical FunctionsFunctions

� xor(x,y) Exclusive OR operation. Return ones where either x or y is nonzero (True). Return zeros where both x and y are zero (False) or both are nonzero (True).

� any(x) Return one if any element in a vector x is nonzero. Return one for each column in a matrix that has nonzero elements.

� all(x) Return one if all elements in a vector x are nonzero. Return one for each column in a matrix x that has all nonzero elements.

TwoTwo--Dimensional GraphicsDimensional Graphics

The plot command through examplesx=linspace(0,2*pi,30); % creates row vector x starting with zero, ending at 2*pi, having

30 elements

y=sin(x);

z=cos(x);

plot(x,y,x,z) % plots vector y vs. vector x and vector z vs. vector x

xlabel('Independent Variable X') % labels the x-axis

ylabel('Dependent Variables Y and Z') % labels the y-axis

title('Sine and Cosine Curves') % entitles the figure


x=linspace(0,2*pi,30); % creates row vector x starting with zero, ending at 2*pi, having 30 elements

y=sin(x);

z=cos(x);

plot(x,y,'b:p',x,z,'c-o') % plots vector y vs. vector x and vector z vs. vector x,

b:p and c-o specify colour, line style and marker of the curve

xlabel('Independent Variable X') % labels the x-axis

ylabel('Dependent Variables Y and Z') % labels the y-axis

title('Sine and Cosine Curves') % entitles the figure

grid on % adds grid lines to the current axes

axis off % turns off all axis labeling and background


Symbol Colour Symbol Marker Symbol Line styley yellow . point - solidm magenta o circle : dottedc cyan x x-mark -. dashdot r red + plus -- dashed g green * starb blue s squarew white d diamondk black v triangle (down)

^ triangle (up)< triangle (left)> triangle (right)p pentagramh hexagram


Other two-dimensional plotting features� loglog is the same as plot, except that logarithmic

scales are used for both axes.� semilogx is the same as plot, except that the x-axis

uses a logarithmic scale, and the y-axis uses a linear scale.

� semilogy is the same as plot, except that the y-axis uses a logarithmic scale, and the x-axis uses a linear scale.

ThreeThree--Dimensional GraphicsDimensional Graphics

Line plots

The plot command from the 2-D world can be extended into three dimensions with plot3. The format is the same as the 2-D plot, except the data are in triples rather than in parts.


Line plots

The plot3 command through an examplet=linspace(0,10*pi); % generates a row vector of 100 linearly

equally spaced points between 0 and 10*pi.

plot3(sin(t),cos(t),t) % where sin(t),cos(t) and t are three vectors of the same length, plot3 plots a line in 3-space through the points whose coordinates are the elements of x, y and z.

title('Helix'),xlabel('sin(t)'),ylabel('cos(t)'),zlabel('t') % labels the x,y,z axis and entitles the figure



Mesh and Surface plots

MATLAB defines a mesh surface by the z

coordinates of points above a rectangular grid in the

x-y plane. It forms a plot by joining adjacent points

with straight lines. The result looks like a fishing net with the knots at the data points. Mesh plots are very useful for visualizing large matrices or for plotting functions of two variables.


Mesh and Surface plotsThe first step in generating the mesh plot of a function of two variables, z=f(x,y), is to generate X and Y matrices consisting of repeated rows and columns, respectively, over some range of the variables x and y. MATLAB provides the function meshgrid for this purpose. [X,Y]=meshgrid(x,y) creates a matrix X whose rows are copies of the vector x, and a matrix Y whose columns are copies of the vector y. This pair of matrices may then be used to evaluate functions of the two variables using MATLAB’s array mathematics features


The mesh and meshgrid command through an example

x=-7.5:.5:7.5;

y=1:31;

[X,Y]=meshgrid(x,y); % X and Y arrays for 3-D plots

R=sqrt(X.^2+Y.^2);

mesh(X,Y,R) % 3-D mesh surface



Mesh and Surface plots

A surface plot of the same matrix R looks like the mesh plot previously generated, except that the spaces between the lines are filled in. Plots of this type are generated using the surf function, which has all of the same arguments as the mesh function.


The mesh and meshgrid command through an example

x=-7.5:.5:7.5;

y=1:31;

[X,Y]=meshgrid(x,y); % X and Y arrays for 3-D plots

R=sqrt(X.^2+Y.^2);

surf(X,Y,R) % 3-D coloured surface


Formulae Booklet for MSP4007 Exam

2015


1

Mathematical Topics

Page No

Algebra 2‐3

Differentiation 4‐6

Integration 7‐8

Complex Numbers 9

Sequences and Series 10

Matrices 11

Binomial Distribution 11

Probability 11


2

Logarithms

log log log

log log log

log log

log 1 and log 1 0

n

a a

A B A B

AA B

B

A n A

a

2 2

22

10 1

Algebra

Formula for quadratic equations

4If 0 then

2

Laws of indices

11

mnm n m n m n m mn

n

m mn nn n

x k x k x k

b b acax bx c x

a

aa a a a a a

a

a a a a a aa


3

22

2

Partial fractions

If the degree of the polynomial is less than that of the denominator

then:

...

f x

f x A B

x a x b x a x b

f x Ax B C

ax bx c dx eax bx c dx e

f x A B

ax b cxax b cx d

2

C

d cx d


4

1

Differentiation

'

, constant 0

, any constant

1log

sin

n n

x x

e

dyy f x f x

dxk

x n nx

e e

Inx xx

x

2

cos

cos - sin

sintan sec

cos

x

x x

xx x

x

'

1

'

Differentiation

Rule 1- Derivative of a constant is zero.

If 4

0

Rule 2 - Derivative of

If then

Rule 3 - Multiplication by a constant

If

that is, the derivative

r

r r

y f x

dyf x

dx

x

dyy x rx

dx

dy k f x kf x

dx

is times the derivative of the function k f x


5

Rule 4- Derivative of sums and differences

If and are two functions of then

Like wise the difference rule

If and are two functions of then

f x g x x

d d df x g x f x g x

dx dx dx

f x g x x

d d df x g x f x g x

dx dx dx

Exponents

exp exp

Alternative notation

e ax ax

dax a ax

dx

dae

dx

The chain rule

Suppose is a function of and itself is a function of .

and

We can differentiate each of these giving

and

To obtain we use the Chain Rule, which st

y z z x

y f z z g x

dy d dz df z g x

dz dz dx dxdy

dx

stes:

dy dy dz dy dz

dx dz dx dz dx


6

The product rule

If and are both functions of

then the derivative of the product

is

u f x v g x x

y u v uv

dy dv duu v

dx dx dx

2

The quotient rule

du dvv udy dx dx

dx v


7

1

1

Integration

, constant

, 1 1

1 0

nn

y f x f x dx F x c

k kx c

xx n c

x

x Inx c xx

In x c x

0

sin -cos

cos sin

tan In sec 2 2

x xe e c

x x c

x x c

x x c x

1

Integration

Rule 1 - Indefinite integral of

If then 1

where is a constant and 1

n

nn

x

xf x x f x dx C

nC n

1

1

Rule 2 - Indefinite integral of

The indefinite integral of is the natural logarithm

1 0

where is an arbitrary constant

x

x In x

dx In x C xx

C


8

Rule 3 - Multiplication by a constant

If is a constant then

k

kf x dx k f x dx

Rule 4 - Integrals of sums and differences

If and are functions of then

If and are functions of then

f x g x x

f x g x dx f x dx g x dx

f x g x x

f x g x dx f x dx g x dx

Rule 5 - Linear composite rule

If and are constants then

1

where

that is, is the indefinite integral of

a b

f ax b dx F ax b Ca

F x f x dx

F x f x

lnk k

dx ax b Cax b a

Integration by parts

dv duu dx uv v dx

dx dx


9

Complex Numbers

Cartesian form: where 1

Polar form:

cos sin

cos , sin , tan

Exponential form:

Euler's relations

cos sin , cos sin

Multiplication an

i

i i

z a bi i

z r i r

ba r b r

a

z re

e i e i

1 11 2 1 2 1 2 1 2

2 2

d division in polar form

,

If , then

De Moivre's theorem

cos sin cos sin

rather than may be used to denote 1

n n

n

z rz z r r

z r

z r z r n

i n i n

j i


10

2

Sequences and Series

Arithmetic progression: , , 2 ,...

first term, common difference

term 1

Sum of terms, S 2 12

Geometric progression: , , ,...

first term, common r

n

a a d a d

a d

kth a k d

nn a n d

a ar ar

a r

1

atio,

term

1Sum of terms, , provided 1

1

when -1 11

k

n

n

kth ar

a rn S r

ra

S rr

2 3

3 5 7

2 4 6

Power series

1 ... for all 1! 2! 3!

sin ... for all 3! 5! 7!

cos 1 ... for all 2! 4! 6!

x x x xe x

x x xx x x

x x xx x


11

11 12 13

21 22 23

31 32 33

22 23 21 23 21 2211 12 13

32 33 31 33 3

Matrices and Determinants

The 2 x 2 matrix has determinant

The 3 x 3 matrix has determinanta

a bA

c d

a bA ad bc

c d

a a a

A a a a

a a a

a a a a a aA a a a

a a a a a

1 32a

Binomial Distribution

!

! !

Where is the number of ways of selecting objects from distinct objects

with the order of selection being unimportant.

Probability

k n k

n n

r r n r

nr n

r

np q

k

is the probability

is the probability of an occurrence happening

is the probability of it not happening

1

In above and are interchangeable

p

q

p q

k r

tw36 university of bolton school of engineering b… · ... automotive performance engineering...

Documents