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Engineering Mathematics UFMFJ9-30-1

Module Handbook 2017 - 2018

Semester One

Contents

Key Module Information 3

Worksheets Topic 1 Dimensions 10

Topic 2 Functions 12

Topic 3 Differentiation 18

Topic 4 Sequences and Series 24

Topic 5 Complex Numbers 27

Topic 6 Integration 30

Worksheet Solutions 35

Formula Sheet 52

1

Key Module Information

1. Module Team

1.1 Module Leader Tim Swift [email protected]

1.2 Lecturers Gary Atkinson [email protected] MATLAB Semester One Basil Norbury [email protected] Mathematics Semesters One / Two Jan Van lent [email protected] Mathematics Semester Two

1.3 Tutors (Mathematics) Tutors (MATLAB) Thom Griffith Gary Atkinson Susan Hawkins Stephen Haynes Stephen Haynes Xiaodong Li

Xiaodong Li Jan Van lent Basil Norbury Emily Walsh

Indunil Sikurajapathi Jan Van lent

2. Teaching and Learning

2.1 Module Learning Materials and Announcements All learning materials - including lecture notes, recordings, worksheets and solutions - can be accessed via Blackboard, as can the regular module announcements. You must get into the habit of checking the module pages on Blackboard at least once a week. At the start of Semester One, please spend some time exploring Blackboard and identifying the location of the various materials.

Worksheets and associated solutions associated with the Semester One Mathematics topics are contained within this handbook. A similar handbook will be available for Semester Two.

2.2 Extra Materials Each week, the lecture notes are cross-referenced to supporting material from the book Mathematics for Engineers (A Modern Interactive Approach) by A.Croft and R.Davison. You are not required to buy this text, but it may be a useful support for your studies.

Another useful resource, also referenced within the notes, is the suite of HELM (Helping Engineers Learn Mathematics) materials, which can be found online at www.uwe.ac.uk/espressomaths. These materials are password protected with username helm and password eMhelm123.

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2.3 Module Content The module syllabus covers a range of tools that are used in Engineering. These tools will be exploited and developed in later modules. The content includes a range of mathematical concepts and techniques and also an introduction to numerical methods and programming using the software MATLAB. 2.4 Mathematics Delivery Each week throughout both semesters you are required to attend two hours of lectures and a one hour tutorial. A set of e-assessment questions will be available every week for you to attempt. These questions are designed to test your understanding of the week’s material and can be attempted as many times as you wish. Each question provides feedback as well as the correct solution. You must make every effort to attempt the e-assessment questions each week and you should raise in your tutorial those questions and points of the week’s lecture that you did not understand. Note that the weekly e-assessments will count towards your coursework assessment mark (for more details, please see the Assessment section below). Further questions on each topic are found in the worksheets, which may be found in this handbook. 2.5 MATLAB Delivery Programming using MATLAB is a skill that employers expect from graduate engineers. In this module, the MATLAB teaching is via a blend of lectures and computer lab’ workshops that take place throughout Semester One. A separate set of notes covering MATLAB will be distributed in the first MATLAB lecture at the start of Semester One. It is essential you attend the computer lab’ workshops to practice MATLAB as there will be a MATLAB assignment at the end of Semester 1. 2.6 Studying on the Module The Mathematics lectures will cover the essential material. You can either take notes in the lecture or listen and take in as much as you can during the lecture, and then make your own notes by referring to the lecture recording and annotated lecture notes posted on Blackboard each week. You will know if you understand something only by doing the corresponding worksheet exercises and e-assessments. Any queries can be raised in your scheduled tutorial in the following week. Points of concern will be covered in the tutorial in order to help you to improve your e-assessment score before the e-assessment engagement cut-off time. In the tutorial, the tutor will also go over concepts and those exercises from the worksheets that students found difficult. Solutions to the worksheets are available in this handbook and on Blackboard. It is essential that you prepare for and attend tutorials. Regarding MATLAB, again the lectures will cover the essential points. You will then have the opportunity to practice using the software in the computer lab’ workshops. The only way to learn MATLAB is through practice and so it is essential that you attend all the workshops in order to work through the MATLAB exercises.

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3. Assessment 3.1 Overview The module has two assessment components, namely the examination (75% of the module mark) and the coursework (25% of the module mark). In order to pass the module, you must achieve a minimum mark of 35% both in the examination and in the coursework, and you must also have an overall module mark of at least 40%. 3.2 Examination Regarding the examination component, there are two elements; the first examination (weighted 30%), comprising a selection of questions from Semester One weekly e-assessments, takes place during the January Assessment period, and the second examination (weighted 70%), based solely on Semester Two material, takes place during the May Assessment period. Each of the examinations is of duration 120 minutes. Note that the first element takes the form of Dewis e-examination, whilst the second is a traditional written examination. 3.3 Coursework The coursework component consists of two elements, namely Dewis e-assessments (weighted 50% of the coursework), which are scheduled throughout both semesters, and a MATLAB assignment (weighted 50% of the coursework), which is based on work in Semester One and submitted early in Semester Two. There are 22 e-assessments, which are scheduled throughout the year (12 in Semester One and 10 in Semester Two). Details regarding access to these assessments may be found under the Assessment button on Blackboard. The e-assessments are run using the Dewis system. This system supplies you with feedback and with your mark immediately after you have submitted your answers. Each e-assessment is worth two marks: the first mark is for engagement and will be given if you attain a mark of at least 40% before the e-assessment engagement cut-off time; the second mark is for attainment and will be proportional to the highest mark scored in the e-assessment. You can do the e-assessment as many times as you like in order to improve your attainment mark. Each of the e-assessments will remain open until the following date: 2018 April 26 / 14:00. Here are some examples of computing e-assessment engagement and attainment marks. (i) You score 42% in e-Assessment 1 by the engagement cut-off time, but your highest

score is 80%, achieved after three more attempts after that date. Your mark for e-Assessment 1 is 1.80.

(ii) You score 42% in e-Assessment 1 by the engagement cut-off time, and you decide to have no more attempts after that date. Your mark for e-Assessment 1 is 1.42.

(iii) You score 30% in e-Assessment 1 by the engagement cut-off time, but your highest score is 65%, achieved after four more attempts after that date. Your mark for e-Assessment 1 is 0.65.

(iv) You do not attempt e-Assessment 1 by the engagement cut-off time, but your highest score is 33%, achieved after two attempts after that date. Your mark for e-Assessment 1 is 0.33.

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Your best 20 e-assessment scores from 22 will be combined to give a mark out of 40, which then will be rescaled to give your percentage mark for the e-assessments element of the coursework. The MATLAB assignment is based on Semester One work and will be submitted online at the start of Semester Two. Details of the assessment will be distributed via Blackboard.

4. Support and Resources 4.1 Peer Assisted Learning (PAL) Each of you has been allocated to a PAL group. The PAL scheme has been a successful innovation at UWE for supporting students. Your PAL group leader is a student who has successfully completed the first year programme. Their rôle is to help you to develop study skills that will help you to be successful and to promote the use of the various learning resources that are available. 4.2 How to Find Help If you encounter difficulties in understanding the material or in answering any of the worksheet questions, then there is a range of support available to you. You could: use those materials on Blackboard that suit your style of learning, e.g., lecture

recordings, annotated lecture notes, the links to HELM, tutorial videos; use the resources available via mathcentre (http://www.mathscentre.ac.uk/); after attempting the worksheet questions and the weekly e-assessments, ask questions

in your tutorial; raise the issue in your next PAL session or email your PAL tutor; use espressoMaths, the Mathematics drop-by station in OneZone (see below). It is highly likely that if you do not understand something, then your peer group is experiencing similar problems. Thus, please ask questions in lectures and, especially, in tutorials. Anybody who is particularly worried about their Mathematics or about their progress on the module should contact the lecturer by email. Please note that you will be contacted by your module tutor if you are not attending tutorials and not engaging with the weekly tests. 4.3 espressoMaths This is a drop-by station in a corner of OneZone (the main refectory) that provides one-to-one Mathematics and Statistics support. The station is staffed Monday to Friday, 12:00-14:00, during teaching weeks. For links to many useful resources, please see the dedicated espressoMaths website at http://www.uwe.ac.uk/espressomaths.

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5. Schedule Here is the week-by-week schedule. Note that CD below refers to the book Mathematics for Engineers (A Modern Interactive Approach) by A.Croft and R.Davison and that HELM refers to sections of HELM (Helping Engineers Learn Mathematics) materials, which can be found at http://www.uwe.ac.uk/espressomaths. 5.1 Semester One

1. Introduction to the Module, Dimensions Supporting notes: CD Chap 6 Block 8.5, HELM 2.7

2. Functions (Linear, Step, Modulus, Quadratic, Polynomials, Rational, Exponential) Supporting notes: CD Chap 7 Block 6, Chap 8 Blocks 1, 3

3. Functions (Trigonometric), Trigonometric Identities, Trigonometric Equations, Waves Supporting notes: CD Chap 9, HELM 4.2, 4.3, 4.5

4. Differentiation (Standard Functions, Higher Derivatives, Product Rule, Quotient Rule, Chain Rule, Parametric, Implicit)

Supporting notes: CD Chap 15 Blocks 1, 2, 3, Chap 16 Block 1, HELM 11.3, 11.4

5. Differentiation (Local/Global Maxima and Minima, Points of Inflection, Partial) Supporting notes: CD Chap 16 Blocks 2, 3, 4, 7, Chap 21 Blocks 1, 2, 3, HELM 11.5,

11.6, 12.2, 18.1, 18.2

6. Sequences and Series (Limits, Arithmetic, Geometric, Binomial Theorem) Supporting notes: CD Chap 19 Blocks 1, 3, HELM 16.1, 16.3

7. Taylor Series and Maclaurin Series Supporting notes: CD Chap 19 Block 4, HELM 16.5 Complex Numbers Supporting notes: CD Chap 11 Blocks 1, 2, 3, 4, HELM 10

8. Complex Numbers Supporting notes: CD Chap 11 Blocks 1, 2, 3, 4, HELM 10

9. Complex Numbers Supporting notes: CD Chap 11 Blocks 1, 2, 3, 4, HELM 10 Integration (Definite Integrals, Trigonometric Functions, Mean Value and Root Mean

Square) Supporting notes: CD Chap 17 Blocks 1, 2, 3, 8 Chap 18 Block 6, HELM 13.1, 13.2,

13.6, 14.2

10. Integration (By Parts, Substitution, Partial Fractions) Supporting notes: CD Chap 17 Blocks 5, 6, 7, HELM 13.4, 13.5, Sections 3, 4

11. Integration Applications (Volumes of Revolution) Supporting notes: CD Chap 18 Blocks 1, 2, 3, HELM 14.1, 14.3

12. Integration Applications (Centre of Mass, Moments of Inertia) Supporting notes: CD Chap 18 Blocks 3, 4, HELM 15.2, 15.3

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5.2 Semester Two

1. Vectors (Magnitude, Addition, Scalar Multiplication, Scalar Product, Projection) Supporting notes: CD Chap 14 Blocks 3, 4, 5, HELM 9.3, 9.4, 9.5

2. Vectors (Vector Product, Applications) Supporting notes: CD Chap 14 Blocks 3, 4, 5, HELM 9.3, 9.4, 9.5 Matrices (Transpose, Addition, Multiplication, Scalar Multiplication) Supporting notes: CD Chap 12 Blocks 1, 2, 3, HELM 7.1, 7.2, 7.3

3. Matrices (Determinants, Inverse Matrices, Matrix Equations, Applications) Supporting notes: CD Chap 12 Blocks 1, 2, 3, HELM 7.1, 7.2, 7.3

4. Solving Linear Algebraic Systems (Gaussian Elimination, Existence and Uniqueness) Supporting notes: CD Chap 12 Block 4, Chap 13 Blocks 2, 3, HELM 8.2, 8.3 Eigenvalues and Eigenvectors (Characteristic Equation, 2x2 Matrices) Supporting notes: CD Chap 14 Blocks 3, 4, 5, Chap 13 Block 4, HELM 9.3, 9.4, 9.5

5. Eigenvalues and Eigenvectors (3x3 Matrices, Cayley-Hamilton Theorem, Diagonalisation of Square Matrices)

Supporting notes: CD Chap 13 Block 4, HELM 22

6. Differential Equations Supporting notes: CD Chap 20, Blocks 1, 2, 3, HELM 19.1, 19.2

7. Differential Equations Supporting notes: CD Chap 20 Blocks 5, 6, HELM 19.3, 19.4

8. Differential Equations, Laplace Transforms Supporting notes: CD Chap 22 Block 1, HELM 20.1, 20.2

9. Laplace Transforms Supporting notes: CD Chap 22 Block 2, HELM 20.2, 20.3

10. Laplace Transforms Supporting notes: CD Chap 22 Block 3, HELM 20.4

11. Examination Preparation See previous examination papers on Blackboard.

12. Examination Preparation See previous examination papers on Blackboard.

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6. Key Dates Here are some key dates that are associated with the module. 6.1 UWE Semesters

Semester One Week 1 (UWE Week 10) / 2017 September 25 Week 12 (UWE Week 21) / 2017 December 15

Semester Two Week 1 (UWE Week 27) / 2018 January 22 Week 12 (UWE Week 40) / 2018 April 27 6.2 UWE Assessment Periods

Assessment Period One 2018 January 08 - January 19

Assessment Period Two 2018 April 30 - May 25 6.3 Coursework Submission Dates

Dewis e-Assessments The Dewis e-assessments are accessible throughout the year until the following date: 2018 April 26 / 14:00. (Note that your attainment mark for each e-assessment is based on the highest score

that you achieve for the e-assessment before this date. After this date, you can still access the e-assessments for revision, but your mark will not count towards the module assessment.)

MATLAB Assignment The submission date for the MATLAB assignment is as follows: 2018 January 25 / 14:00.

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Engineering mathematics UFMFJ9‐30‐1 Worksheet 1 : Dimensions

1. Give the dimensions in terms of the fundamental dimensional quantities mass [M], time [T] and length [L] :

(a) velocity,

(b) acceleration, (c) force, F

(d) pressure , , (force/unit area)

(e) density, mass/volume

(f) stress, force per unit area

(g) surface tension,

(h) viscosity Measure of

resistance to fluid flow, . SI units

(i) curvature,

2. Check using dimensions the validity of the following statements

(a) F mgl (b) 21

2s ut at (c) 2 2 2v u gs

where m = mass, g = acceleration due to gravity, l = length, t = time, u and v = velocities, s = distance, F = force

3. Check that the following formulae define non‐dimensional quantities

(a) Reynolds Number vl

(b) Mach Number v

c (c) Euler Number

2

p

v

(d) Froude Number v

gl (e) Weber Number

2v l

where = velocity, = acceleration, = length, = speed of sound, = pressure, = density, μ = viscosity, σ = surface tension, same units as force per unit length.].

4. The strings of a musical instrument are metal wires pulled taught. The frequency f of

vibration of any string is given by

cbalkf

where is the tension in the string, l is its length, its mass per unit length and k

is a non‐dimensional constant. Noting that 1][][ Tf , give the values for a, b and c.

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Engineering mathematics UFMFJ9‐30‐1 Worksheet 1 : Dimensions

5. (Question from MJD Barry)

The velocity of a high speed test vehicle approaching maximum speed satisfies the equation

bxBeb

av

where a, b and B are positive constants and x is the distance travelled from the start point. Identify the dimensions of a, b and B.

Extra Question E1.

(i) Find the dimensions of , pressure, given that the Euler number,

is a dimensionless number. ( is the density, is velocity).

(ii) Check if the following equation is dimensionally valid

where is energy, is density, is acceleration due to gravity, is height. (Note the SI units of energy are .)

(iii) The velocity of a parachutist is given by

where is time and , , and are constants. Find the dimensions of , , and .

11

Engineering Mathematics

UFMFJ9-30-1 Worksheet 2 : Functions

1

Straight lines and quadratics

1. Solve the flowing equations (by whatever method is suitable):

(a) 𝑥2 + 5𝑥 − 6 = 0 (b) 𝑥2 − 7𝑥 − 30 = 0 (c) 𝑥2 + 6𝑥 + 1 = 0

(d) 3𝑥2 − 𝑥 − 1 = 0 (e) 𝑥3 − 9𝑥 = 0

2. Sketch the following:

(a) 𝑢(𝑡 − 1) (b) 𝑢(𝑡 + 1) − 𝑢(𝑡 − 1) (c) |𝑥 − 2|

3. Complete the square for the following expressions and sketch their graphs:

(a) 𝑥2 + 4𝑥 + 3 (b) 𝑥2 + 4𝑥 + 4 (c) 𝑥2 + 4𝑥 + 10

4. Complete the square for the following expressions:

(a) 𝑥2 − 3𝑥 + 1 (b) 2𝑥2 − 6𝑥 + 7

Partial Fractions

5. Identify the poles and zeros of the following rational functions:

(a). 2

1

xy (b)

23

42

xx

y (c) 1

42

2

x

xy (d)

2

12

x

xy

6. Express each of the following as the sum of its partial fractions:

(a) )2)(1(

15

xx

x (b) 23

32 xx

(c) )2)(1(

32 xx

Exponentials 7. Consider the functions

(a) 1 , 0xy e x (b) 4

, 01 t

y te

Sketch a graph of the function and deduce the range of values that the function can take from the graph

12



2

8 . Solve the following equations

(a) 42 xe (b) 21

1

xe (c) 23 te

(d) 2

12

te (e) ln( 3) 2x (f) 2)ln( 2 x

9. Solve the equation 2 3(3 2) 0tt e , 0t

10. The temperature, T, of a chemical reaction is given by 0,120 02.0 teT t .

Calculate the time (measured in seconds) needed for the temperature to double

its initial value.

11. The temperature T (°C)of a cooling body at time t (minutes) is given by

teT 4.08515 .

a) Sketch a graph to show how the temperature varies over time.

b) State the initial temperature and the temperature of the surroundings.

c) Calculate when the temperature reaches 50°C.

12. The current I in an electrical circuit is given by

t

L

R

eR

EI 1 where E, R and L are

constants. Determine (a) the steady state current in the circuit, (b) the time taken for the

current to reach 98% of its steady state value in terms of R and L.

13



3

Trigonometry

13. Show that sin(𝑡) sec(𝑡) = tan(𝑡)

14. Express the following as the sum (or difference) of two sines (or cosines)

(a) sin(5𝑥) cos(2𝑥) (b) 8 cos(6𝑥) cos(4𝑥) (c) 1

3sin (

1

2𝑥) cos (

3

2𝑥)

15. Show that (a)

sin(4𝑥) + sin(2𝑥)

cos(4𝑥) + cos(2𝑥)= tan(3𝑥)

(b)

√2 − 2 cos(6𝑡) = |2 sin(3𝑡)| 16. Find all the solutions of the following trigonometric equations in the specified interval

(a) cos(𝑥) =1

2 , 0 ≤ 𝑥 ≤ 2𝜋 (b) sin(𝑥) = −

√3

2,0 ≤ 𝑥 ≤ 2𝜋

(c) sin 0.8t , 0 2t (d) 4cos 2t , 0 2t

(e) tan 2 4,t 0 t (f) sin(2 ) 0.4, 0 2t t

17. The figure below shows the oscilloscope trace of a sine wave oscillating about a non-zero mean value. State the equation of the wave

18. Write down the amplitude, period and time shift of:

(a) 𝑦 = 3sin(2𝑡 −𝜋

3) (b) 𝑦 = 15 cos (5𝑡 −

3𝜋

2)

y

t

14



4

19. The Fahrenheit temperature at a certain location over one complete day is modelled by

𝐹(𝑡) = 60 + 10 sin (𝜋

12(𝑡 − 8)) 0 ≤ 𝑡 ≤ 24

where t is measured in hours.

(a) What are the temperatures at 8am and 12 noon

(b) At what time is the temperature 600F

(c) Obtain the maximum and minimum temperatures and the time at which they occur.

20. Express the following in the form 𝑅 sin(𝑤𝑡 + 𝛼)

(a) 𝑦 = −√3 sin(2𝑡) + cos(2𝑡) (b) 𝑦 = cos(2𝑡) +√3 sin(2t)

Extra questions

E1: (a) Determine the poles and zeros of the rational function

𝑦 =3𝑥 + 4

𝑥2 + 𝑥 − 12

[3 marks]

(b) Consider the function

𝑓(𝑡) = 1 + 𝑒−𝑡,𝑡 ≥ 0

State the range of 𝑓(𝑡) and sketch the function. [3 marks]

(c) Rearrange 𝑥2 + 10𝑥 + 30

by completing the square and hence sketch 𝑥2 + 10𝑥 + 30, clearly showing the coordinates at its minimum value.

[4 marks]

(c) Solve the following equations: 1

1 − 𝑒𝑥= 4

[2 marks]

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5

E2: (a) Find the poles and zeros of the function 2

3 1

2 15

xf x

x x

[3 marks]

(b) Expand the following algebraic fractions in terms of their partial fractions:

(i) 31

62

xx

x

[3 marks]

(ii) 21

52 xx

x

[4 marks]

Trigonometry

E3: (a) Solve the following equations in the specified intervals:

(i) tt 0 ,75.03cos [4 marks]

(ii) zzne z 025 [3 marks]

(b) Write down an equation for this graph. All numerical values used should be integers.

[3 marks]

(c) Write the expression 2cos 4 5sin 4t t as a single sinusoidal expression of the

form sin( )A kt , where , and A k are constants that need to be determined.

[4 marks]

16



6

E4: (a) Solve the following equations:

(i) cos(t) + cos(2t) = 0, 0 ≤ t ≤ π [2 marks]

(ii) cos(2𝑡) = 1

2,0 ≤ 𝑡 ≤ 2𝜋

[4 marks]

(b) Write the expression 3 cos(3𝑡) + 2 sin(3𝑡)

as a single wave of the form 𝐴 cos(𝑘𝑡−∝)

Hence give the maximum value for

6 + 3 cos(3𝑡) + 2 sin(3𝑡)

and the values for 𝑡, 0 ≤ 𝑡 ≤ 2𝜋 at which the maximum occurs.

[8 marks]

17


UFMFJ9-30-1 Worksheet 3 : Differentiation

1

Differentiation

1. Find dx

dy when y is given by

(a) ( )xxx sin203023

++−

(b) 5232

+−+−− xxx

eee

(c) ( )xx

x31

ln3 −+

(d) ( ) ( )xx 3sin42cos3 −

(e) xx

+

5tan2

2. Calculate the rate of change of each function when 6.1=t . State whether the function is

increasing or decreasing.

(a) 23532

+− tt

(b) ttee

2

3

2 −−−

(c)

+

2

3cos

2sin3

tt

(d) 2

3ln2

tt −

3. For each of the problems defined in Q1, find 2

2

dx

yd.

4. Determine the equation of the tangent line for each of the following functions at the

specified point.

(a) 252

−+= xxy at 2=x

(b) xey

−+= 1 at 1=x

18



2

Product Rule, Chain Rule

5. Using the product rule, differentiate the following

(a) ( )xeyx

3cos2−

=

(b) ( ) ( )xxy 2sin532

−=

(c) ( ) xexy

21−=

6. The displacement s of a particle is given by t

tes5.0

4−

= . Find the value of t at which the

speed of the particle is equal to zero.

7. Using the chain rule, differentiate the following

(a) 32

+= xy

(b) ( )xy 2sin34

=

(c) 2/2

xey

−=

8. Decide which rule (product, quotient or chain) you need in order to differentiate each function

and hence find the rate of change of y at the specified point:

(a) ( ) ( )26 ln 3 , 1y x x x= − =

(b) , 2x

y e x−

= =

(c) ( )1 , 2x

y x e x−

= − =

Parametric differentiation and Implicit differentiation

9. For each of the following parametrically defined functions find dy

dx[leave your answer in terms

of t]

(a) 21, 2x t y t= + =

(b) ( ) ( )sin 2 , cos 2x t y t= =

19



3

(c) ( ) ( )sin 2 , cos 3x t y t= =

(d) 2,

tx t y e

−= =

10. Find dy

dxfor each of the following implicit functions, given that y is a function of x .

(a) 5 ln� + �� = sin 3� − �

(b) 9�� + 2�� = 8

(c) � ln� + �� = 6��

Optimisation

11. Find and identify any turning points of the following functions

(a)3 21

2 5 43

y x x x= + − −

(b)3 2

4 2y x x x= − − +

(c) ( )23

xy x e

−= −

12. An object is projected vertically upwards. Its height above ground level at time t is given by

28.950 tth −=

Determine the time at which the speed of the object is instantaneously zero.

13. The angular displacement of a rotating cylinder is given by

tt sin3cos2 +=θ

Find expressions for the angular speed and the angular acceleration of the cylinder. At what

time is the angular speed first zero? [Mustoe]

14. Identify the maximum and minimum values of each of the following functions in the specified

interval.

(a) 24 , 2 4y x x= − − ≤ ≤

(b) , 0 2x

y e x= ≤ ≤

20



4

(c) 3 23 2 1 , 0 3y x x x x= − + + ≤ ≤

15. A beam of length l and weight w per unit length is clamped horizontally at both ends. Its

deflection y at a distance x from one end is given by

( )2

2

24

wxy l x

EI= −

where E and I are constants. Show that the maximum deflection of the beam is

4

384

wl

EI

[Modified from Evans]

16. The power P transmitted by a belt drive is given by the function 3

P Tv cv= − where T is the

tension in the belt, v is the speed of the belt and c is a positive constant. Show that the

maximum power transmitted by the belt occurs when c

Tv3

= . Determine an expression for

the maximum power generated by the drive belt.

Partial Differentiation

17. Find the first derivatives of the following functions

(a) � = 5�� − 3�� − 2 (b) � = 2�� + 6� − sin(2�)

(c) � = sin(��) (d) � = (�� + ��)��

18. Find the two first derivatives and the four second derivatives of the following functions, and,

in each case, confirm that ��

��=

��

��.

(a) � = 2�� + 5�� − 8�� (b) � = 2�� + 4� − 5

(c) � = 2��cos(6�) (d) � = 2�" + sin(��) − 3ln(�)

(e) � =#

�ln(�� + ��) (f) � = ��

(g) � =�

#$��$� (h) � = ��sin(�)

21



5

Extra questions

E1: (a) For each of the following functions, determine dx

dy

(i) � = cos(��)

[3 marks]

(ii) � = �"cos(2�)

[3 marks]

(b) A function is defined parametrically by

� = cos(2%) ,� = %� + 1

Evaluate dx

dy when % =

(

�

[4 marks]

(c) The force between two particles of mass ) and * respectively is given by

+ = −,)*

-�

where - is the distance between the two particles and , is a constant.

Given that initially - = . and the particles are moving towards each

other at a constant rate of 1m/s, obtain the rate of change of + when -

is half its initial value.

[4 marks]

E2: (a) Find dy

dx for each of the following functions:

(i) ( ) ( )xxy tansin=

[2 marks]

(ii) ( )( )siny n x= l

[2 marks]

(iii) � = (�� − 5)��

[3 marks]

(b) A particle is moving such that, in terms of the time elapsed t , the x- and y-

coordinates are given by:

22



6

6tx t e−

= + 2t

y t e−

= +

Determine dy

dx in terms of t and explain what is happening as t → ∞ . What does

this mean about the path of the particle?

[4 marks]

(c) Show that if

�� = 1 − ��

then /�

/�= −

�

�

[3 marks]

E3: (a) Find /�//� and /��//�� and hence determine the location and nature of

any turning points of the functions:

(i)� = (��−3)��

(ii)� = 4 + � +4

�

[8 marks]

(b) The torque of an electric machine is given by

1 = 2 cos(2%) + 6 sin(%) ,0 ≤ % ≤ 4

Find /1//%

Determine the time at which /1//% is first zero.

[6 marks]

23

Worksheet 4: Sequences and Series

(1) Determine the limit (if it exists) of each of the following sequences whose

kth term is:

(a) 106/2k (b) 5(1.01)k (c) (−1)k+1 (d) 4k + 3/k

(e) 1−4k3k2+5k (f) 5−8k

6k−17 (g) 7

3+23√k4

(h) 3k2+51−4k+2k2

(2) Determine the limit of 31−4ak as k →∞ if

(a) a = 0.9, (b) a = 1, (c) a = 1.1.

(3) Evaluate the following sums

(a)∑4

k=0|k−2|k! Note: 0! = 1

(b)∑3

k=−1(−1)k+1

(k+2)2

(4) Express the series below in sigma notation

(a) 1 + 2 + 3 + · · ·+ 49 + 50

(b) 3− 3(0.2) + 3(0.2)2 − 3(0.2)3 + · · ·and find the sum of each series.

(5) Write out the first four nonzero terms of

f(t) =∞∑

n=1

1− (−1)n

2nsin(nπt)

(6) Express the series below in sigma notation, starting the summation with

k = 0.

(a) 1/2 + 1/4 + 1/6 + · · ·+ 1/1000

(b) 1− 1/2 + 1/4− 1/8 + 1/16 + · · ·

(c) 1− x+ x2/2!− x3/3! + x4/4! + · · ·

(d) x+ x3/2! + x5/4! + x7/6! + · · ·

(7) Use the binomial theorem to expand

(a) (1 + x)4 (b)(1− y

2

)3(c) (3x− 2y)4

(8) Obtain the first four terms in the expansion of

(a)√

1− x (b) 11+2x

In each case state the range of values for which the expansion is valid.

(9) Use the standard Maclaurin expansions to obtain the first four non-zero

terms in the power series expansion of

(a) e−2x (b) cos(3x) (c) ln(1− 2x) (d) ex2

(e) sin(x/4)

In each case give the range of validity of your expansion.

(10) Use the series found in (8)(a) to find an approximation to√

0.88 and hence

find the error involved in the calculation.

1

24

2

(11) The current I in a circuit is given by I = E(1 − exp(−Rt/L))/R. Write

down the first three terms in the power series expansion of exp(−Rt/L)

and deduce the value of I as R→ 0.

(12) Determine the first four terms in the Taylor series of lnx expanded about

x = 1 and use your result to calculate an approximate value for ln(1.1).

Extra Questions

E1: (a) Determine the limit (if it exists), as k → ∞, of each of the following

sequences, whose kth term is:

(i) (−1)k

(ii) 4+5k1−k

(iii) y = 10(1.01)k

[2 marks]

(b) Write out the first four nonzero terms of

f(t) = 2 +∞∑

n=1

8

nπsin(nπ

2

)cosnπt

[2 marks]

(c) Find the first three terms of the series of sin(3x) expanded about x = 0

and use your result to calculate an approximate value for sin(0.3)

[3 marks]

(d) Find the first three non-zero terms of the Taylor series of sin(x) ex-

panded about x = π/2 and use your result to calculate an approximate

value for sin(1.5).

[3 marks]

E2: (a) If a sequence is defined by the expression

ak =2k − 1

1− 4k

write down the first five terms of the sequence, where the first term is

defined by k = 1.

[1 mark]

(b) Use the Binomial theorem to expand (x− 2y)5.

[2 marks]

(c) Find the sum of the first 500 natural numbers, that is compute

1 + 2 + 3 · · · 499 + 500

[1 mark]

(d) Find the sum of the series

2 + 2(0.8) + 2(0.8)2 + 2(0.8)3 + · · ·

[1 mark]

25

3

(e) Use standard MacLaurin expansions to obtain the first four non-zero

terms in the power series expansion of y = ln(1 − 4x). Use the series

to find an approximation to ln(0.6) and hence find the absolute error

involved in the calculation.

[3 marks]

(f) Determine the first two non-zero terms of the Taylor series expansion

of f(x) =√x about x = 16, and express the resulting expression in

the simplest form possible.

[2 marks]

E3: (a) Evaluate5∑

n=1

|n− 3|n!

giving your answer as a fraction.

[2 marks]

(b) Consider the limit as k →∞ for each of the following expressions. In

each case, state whether or not a limit exists. If a limit does exist then

determine its value.

(i) 12k

(ii) k−11−4k

(iii) cos k

(iv) cos kk

[3 marks]

(c) Use standard MacLaurin expansions to obtain the first four non-zero

terms in the power series expansion of y = sin2 2x. Use the series to

find an approximation to sin2(0.5) and hence find the absolute error

involved in the calculation.

Hint: You may wish to make use of the result sin2 θ = 12 (1− cos 2θ).

[3 marks]

(d) Determine the first two non-zero terms of the Taylor series expansion

of f(x) = 1x about x = 1, and express the resulting expression in the

simplest form possible.

[2 marks]

26

Engineering Mathematics Complex Numbers UFMFJ9-30-1 Worksheet 5

Worksheet 5: Complex Numbers Standard Questions

1. Solve the following quadratic equations

(a) 0162

=+x (b) 0132

=++ xx

2. Determine a quadratic function that has zeros located at 4x j= + and

4x j= − .

3. Determine a cubic function that has zeros located at 2, 1 3x x j= = − +

and 1 3x j= − −

4. Express the following numbers in the rectangular form bjaz += ,

where a and b, are real numbers.

(a) 9−=z (b) 43 −+=z

5. Plot jz +−= 2 together with its complex conjugate on the complex

plane.

6. Simplify the following complex expressions and write them in

Rectangular form bjaz += . Plot each number on the complex plane.

(a) 2j (b) )4)(62( jj +−+ (c)

j

1 (d)

j

j

+

+

3

2

(e) j

j

65

4124

+−

− (f)

j

j

84

65

+−

−

7. Express the following complex numbers in the polar form θ∠= rz with

πθπ ≤<− . Hint: plot the numbers on the complex plane first. (a) 4 5z j= + (b) jz −= 3 (c) jz 32 +−= (d) 4 7z j= − −

27


8. Let 1x and

2x be complex solutions of the quadratic equation

01222

=++ xx

Express both 1x and

2x in the polar form θ∠r .

9. Express the following complex numbers in the rectangular form

bjaz += . θ is given in radians.

(a) 3/2 π∠=z (b) 4/6 π−∠=z (c) πjez 4= (d)

2/3

πjez −=

10. Use the polar form to simplify the following complex expressions. Express your final answer in rectangular form.

(a) 4

3

)34(

)23(

j

j

+

−

(b) 6( 2 )j j− +

(c) 4

(1 4 )j

j

+

11. Using Euler’s formula, show that

2cos

θθ

θjj

ee−

+= and

j

eejj

2sin

θθ

θ−

−=

Extra questions E1

(a) Plot the following complex numbers on the complex plane

(i) jz 32 −=

(ii) 2

3 πz ∠=

(b) Let 1x and

2x be complex solutions of the quadratic equation

01042

=+− xx

Express both 1x and

2x in the polar form θ∠r .

(c) Given 4/21 π∠=z and 5/52 π∠=z , find the real and imaginary parts

of the following complex number 22

31 zz

28


E2.

(a) Carry out the following operations involving complex numbers

(i) Convert the complex number 8 3z j= + to the polar form θ∠= rz ,

where π θ π− < ≤ .

(ii) Convert the complex number 2 / 3z π= ∠ , where the argument of z has been expressed in radians, to the rectangular form jbaz += .

(b) Given 1

4 2z j= − + and 2

3 4z j= − convert both numbers into polar

form, and hence find the real and imaginary parts of the complex

number

5

1

2

z

z

.

E3.

(a) Express the complex number 5

1

jin both rectangular and polar form.

(b) Determine the cubic function that has zeros located at 3x = , 2 3x j= +

and 2 3x j= − .

(c) Given 0.2

14

jw e= , 1.4

22

jw e= and 0.5

3

jw e

−=

Determine the magnitude and the argument of the following complex

numbers

(i) 1

1

w

(ii) 2

1

2

w

w

(iii) 1 2 3

w w w

29


UFMFJ9-30-1 Worksheet 6 : Integration

1

Integration

1. Using the tables of standard integrals find

(a) ( )∫ +−− dxxxx 3523

(b) ∫

+ dxx

x3

17

(c) ( )( )∫ −−

dxxex 2sin46 2

2. Using the tables of standard integrals find

a) ( )∫ −

2/

0

sincos3

π

dxxx (b) ∫−

2

1

3

2 1dt

t

t

(c) ( )∫−

−

0

2

2 dxxe x (d) ∫

1

05

sin dtT

tnπ

3. Find

(a) ∫4/

0

sincos3

π

dxxx (b) ( ) ( )∫2

1

2sin2/sin dttt ππ (c) ∫

1

0

2

4sin dt

T

tnπ

Average Value and Root Mean Square

4. Find the average value of the following functions

(a) ty sin= across the interval π≤≤ t0

(b) tteey

−+= across the interval 10 ≤≤ t

(c) 21 ty += across the interval 31 ≤≤− t

(d) ty2cos= across the interval 2/0 π≤≤ t

30



2

5. Determine the root mean square values of the following functions over the specified interval

(a) y t= over the interval [0, 2]

(b) 2 3 2y t t= − + over the interval [0, 2]

(c) cos(2 )y t= over the interval [0, 2 ]π

(d) sin( )y kt= over the interval [0, 2 ]π , where k is a positive integer.

Integration by Parts, by Substitution, Integrals leading to Logarithms and using Partial Fractions

6. Evaluate the following integrals

(a)

/ 2

0

cos( )x nx dx

π

∫ where n is an integer

(b) 2

/ 2

cos( )x nx dx

π

π

∫ where n is an integer

(c) ∫−

2

0

22dtet

t

7. Evaluate the following integrals

(a) ∫ +

2

042

3dt

t

(b) ∫π

dxxx0

2)sin(

(c) ∫ −

2

1

3

2

12dt

t

t

(d) ∫e

dxx

x

1

)ln(

31



3

8. Integrate the following functions

a)� = � + 1� + 4)2� + 5)

b) 2

( 4)(4 1)( 1)

xy

x x x=

+ + +

c) 2

2

1

( 2)

xy

x x

+=

+

d) 2

2

1

( 1)

x xy

x x

+ +=

+

Applications of Integration

9. (a) A vessel is formed by rotating the curve y t= between 0t = and 2t = through 2π

radians about the t-axis. Determine the volume V of the vessel .

(b) A vessel is formed by rotating the curve siny t= between 0t = and t π= through 2π

radians about the t-axis. Determine the volume V of the vessel.

(Note the volume of revolution is given that ∫=

b

a

dtyV2π )

10. Find the centre of mass of a lamina of constant density bounded by

� = 4�, ��0 ≤ � ≤ 9.

11. Find the moment of inertia of a square lamina of side 2b which is of uniform density, and

mass M , about one of its sides.

12. Calculate the moment of inertia of a uniform rod of mass M and length � about a

perpendicular axis of rotation

(a) at its end.

(b) at its centre

32



4

Extra questions

E1: (a) Determine the mean value of the function

2 + � + ��

over the interval 1 ≤ � ≤ 1.5

[2 marks]

(b) Express as a partial fraction,

� + 4� + 4� − 12

Hence evaluate

� � + 4� + 4� − 12 ��

�

�

[6 marks]

(c) Integrate the following functions with respect to t.

(i) sin 2�)

[3 marks]

(ii) � sin2�)

[3 marks]

E2: (a) Integrate the following functions with respect to t:

(i) ( )2

1sin 4y t

t= + − [3 marks]

(ii) ( )2cosy t= [3 marks]

(b) Find the r.m.s. value of the function ( )1 siny x= + over the interval 02

xπ

≤ ≤ .

[3 marks]

(c) Find the moment of inertial of a thin rod of mass #, and length �, about a

perpendicular axis of rotation that is �/3 from one end.

[5 marks]

33



5

E3 (a) Express sin4�) cos2�) in the form of ( sin)�) + *sin+�).

Hence evaluate

, 6 sin4�) cos2�)��./ / .

(7 marks)

(b) Find the moment of inertia of a square lamina with uniform density of mass M and side

2b about one of its sides.

(7 marks)

34

Engineering Mathematics UFMFJ9‐30‐1 Worksheet 1 Answers

Answers

1. (a) (b) (c) (d)

(e) (f) (g) (h)

2. (b) and (c) are valid dimensionally

3. A dimensional analysis of each ratio should return a value of 1.

4. Answer is2

1,1,

2

1 cba

2/1

l

kf

5. [B] = [LT‐1], [b] = [L‐1], [a] = [T‐1]

Exam type question

35

Engineering Mathematics Worksheet 2: Functions UFMFJ9-30-1 Solutions

1

1. (a) (b) (c)

(d) (e)

2.

(a) (b) (c)

3.

(a) ) ( (b) (c)

4. (a)

(b)

5. (a) No zeros, pole at x=2 (b) No zeros, poles at x—2,-1. (c) zeros at x=-2,2, poles at x=-1,1.

(d) zeros at x=-1,1, pole at x=-2.

6. (a) 2

3

1

2

xx (b)

2

3

1

3

xx (c)

2

3 3 6

5( 2) 5( 1)

x

x x

(d)

2

11

1

1

sss

7.

36


2

1.

(a) range 2 1y (b) range 2 4y

8. All answers given to 2dp (a) 0.69 (b) -0.69 (c) 3.69 (d) 0.83,-0.83 (e) 4.39 (f) 2.72,-2.72

9. 2

3 [Hint: what can you say about the value of te 3 ?]

10. 34.66 seconds (2dp) [= 02.02 n ]

11. a)

b) Initially at 100°C, cooloing to the surrounding temperature of 15°C.

c) 2.22 minutes [= 4.08535 n ; look at the graph to get a first estimate]

12. (a) ,E

t IR

(b) ln(0.02)L

R

13.

0

20

40

60

80

100

120

0 2 4 6 8 10

T (°

C)

t (minutes)

37


3

14. (a)

(b)

(c)

15. (a)

(b)

16. All answers in radians. (Remember rad = 180o.)

(a)

(b)

c) Let )8.0(sin 10

t

since the sine function is +ve

00 or tttt

21.2or 93.0 tt (2dp)

d) 21cos t

Let )(cos211

0t

Since the cosine function is -ve

00 or tttt

19.4or 09.2 tt (2dp)

(e) Let tu 2 so eqn becomes 4tan u

38


4

Let )4(tan 10

u

Since tan function is +ve

00 or uuuu

Hence

2or

2

00 ut

ut

23.2or 66.0 tt (2dp)

(f) Let tu 2 so eqn becomes 4.0sin u

Let )4.0(sin 10

u . Since sine function is +ve

kuukuu 2or 2 00

Hence

2

2or

2

2 00 kut

kut

(the integer k picks out the different cycles of the sine function.)

1st cycle 0k 2nd cycle 1k

37.1or 21.0 tt (2dp) 51.4or 35.3 tt (2dp)

17. Sine wave of amplitude 2.5. There are 3 periods with 2 units. Therefore angular frequency is 3.

The equation oscillates around a mean value of 1.

Therefore the equation of the wave is

18. (a) Amplitude 3, period

, time shift

compared with

(b) Ampitude 15, period

, time shift

compared with

39


5

19. (a)At t=8, temp =60 F. At t=12. temp =

(b) When

In one day, at t=8,(8am) and t=20, (8pm)

(c) Max temp, 70, when sine wave has value 1,

ie when

,

Min temp, 50, when sine wave has value -1,

ie when

,

20. (a) ,

so in 2nd quadrant,

(b)

so in 1st quadrant,

,

40

Engineering Mathematics Worksheet 3: Differentiation UFMFJ9-30-1 Solutions

1

C&D refers to Croft and Davison Mathematics for Engineers Third Edition.

1. (a) 2 390 40 cosdy

x x xdx

(b) 22 3 2x x xdye e e

dx

(c) 2 3 2

3 1 3

2

dy

dx x x x

(d) 6sin 2 12cos 3dy

x xdx

(e) 2

1 2

2 1sec

5 5 2

dy x

dx x

If you have difficulty with Qu 3, have a go at the exercises on C&D pages 706 and 708

2. (a) 210 9dy

t tdt

so at t=1.6, 7.04dy

dt (decreasing)

(b) 222

3

t tdye e

dt

so at t=1.6, 0.053dy

dt (decreasing)

(c) 3 3 3

cos sin2 2 2 2

dy t t

dt

so at t=1.6, 0.032

dy

dt (increasing)

(d) 3

2 6dy

dt t t so at t=1.6, 2.715

dy

dt (increasing)

If you have difficulty with Qu 4, have a go at the exercises on C&D page 710.

3. (a) 2

4

2180 120 sin

d yx x x

dx

(b) 2

2

24 3 2x x xd ye e e

dx

(c) 2

2 2 3 5 2

3 2 9

4

d y

dx x x x (d)

2

212cos 2 36sin 3

d yx x

dx

(e) 2

2

2 3 2

4 1sec tan

25 5 5 4

d y x x

dx x

4. (a) 9 6y x

(b) 1 2

1y xe e

41


2

If you have difficulty with Qu 7, try the exercise on C&D page 747.

5. (a). 2 2cos 3 3sin 3xdye x x

dx

, (b). 26 sin 2 2 3 5 cos 2dy

x x x xdx

(c). 21 2 xdyx x e

dx

If you had problems with Qu 1, go on and try the exercise on p721 of C&D

6. 2t

7. (a). 2 3

dy x

dx x

, (b). 324cos 2 sin 2

dyx x

dx , (c)

2 / 2xdyx e

dx

If you had problems with Qu 4, go on and try the exercises on p729 of C&D

8. (a). [product rule], 2 6

2 ln 3 , 2.803xdy

x xdx x

(b). [chain rule], 1

, 0.085952

xdye

dx x

(c). [product rule], 2 , 0xdyx e

dx

9. (a) 1dy

dx t , (b) tan 2

dyt

dx , (c)

3sin 3

2cos 2

tdy

dx t , (d)

2

tdy e

dx t

If you had problems with Qu 7, go on to try the exercise on p739 of C&D

10.

42


3

11. (a). 1, 6.667 min, 5,29.333 max (b). 2.786, 10.209 min,

0.120, 2.061 max

(c). 1, 5.437 min, 3,0.299 max (d). 4.732, 7.464 min, 1.268,0.536 max

If you had problems with Qu 1, go on to try C&D exercises on p762

12. t = 2.55 seconds (2dp)

13. 2sin 3cosd

t tdt

2

22cos 3sin

dt t

dt

0d

dt

when tan 1.5t , which first happens when 0.98t seconds (2dp)

14. (a).max occurs at 0, 4 , min occurs at 4, 12

(b). min occurs at 0, 1 , max occurs at 2, 7.389

(c). min occurs at 1.577, 0.615 , max occurs at 3, 7

There is nothing similar in C&D. You need to realise that, when you are restricted to an interval on a

graph, the maximum or minimum point may actually be at one of the ends of the interval.

15. First show that maximum deflection occurs at 2

lx and then use this value to determine

the maximum deflection.

16. A sketch of the function 3 2P Tv cv v T cv should confirm that the power takes its

maximum value at a turning point located between 0v and T

vc

.

Next step is to solve 03

dP Tv

dv c . Note that the speed of the belt must be

positive. Finally, substituting 3

Tv

c into 3P Tv cv should yield

3 / 2

max

2

3

TP

c

.

43

Question 17

(a) ux = 5y2, uy = 10xy − 18y5 .

(b) ux = −10e−5x, uy = 6− 2 cos(2y) .

(c) ux = y cos(xy), uy = x cos(xy) .

(d) ux = (x2 + 2x+ y2)ex, uy = 2yex .

Question 18

(a) ux = 8x3 + 5y2, uy = 10xy − 48y5, uxx = 24x2,

uyy = 10x− 240y4, uxy = uyx = 10y .

(b) ux = 2e−y + 4, uy = −2xe−y, uxx = 0, uyy = 2xe−y, uxy = uyx = −2e−y .

(c) ux = −10e−5x cos(6y), uy = −12e−5x sin(6y), uxx = 50e−5x cos(6y),

uyy = −72e−5x cos(6y), uxy = uyx = 60e−5x sin(6y) .

(d) ux = 6x2 + y cos(xy), uy = x cos(xy)− 3/y, uxx = 12x− y2 sin(xy),

uyy = −x2 sin(xy) + 3/y2, uxy = uyx = cos(xy)− xy sin(xy) .

(e) ux = xx2+y2 , uy = y

x2+y2 , uxx = y2−x2

(x2+y2)2 , uyy = x2−y2

(x2+y2)2 ,

uxy = uyx = − 2xy(x2+y2)2 .

(f) ux = y(1− 2x)e−2x, uy = xe−2x, uxx = 4y(x− 1)e−2x, uyy = 0,

uxy = uyx = (1− 2x)e−2x .

(g) ux = − y5(x−2y)2 , uy = x

5(x−2y)2 , uxx = 2y5(x−2y)3 , uyy = 4x

5(x−2y)3 ,

uxy = uyx = − x+2y5(x−2y)3 .

(h) ux = (1+xy) sin(y)exy, uy = x(x sin(y)+cos(y))exy, uxx = y(2+xy) sin(y)exy,

uyy = x(x2 sin(y) + 2x cos(y)− sin(y))exy,

uxy = (2x sin(y) + cos(y) + x2y sin(y) + xy cos(y))exy.

1

44

Engineering MathematicsUFMFJ9-30-1

Worksheet 4Solutions

Answers to Worksheet 4: Series

1. (a) 0 (b) DNE (c) DNE (d) DNE

(e) 0 (f) −43

(g) 0 (h) 32

2. (a) 3, (b) −1, (c) 0.

3. (a) 3.25, (b) 0.8386

4. (a)50∑

k=1

k = 1275, (b)∞∑

k=0

3(−0.2)k = 2.5,

5. f(t) = sin(¼t) + 13sin(3¼t) + 1

5sin(5¼t) + 1

7sin(7¼t) + ⋅ ⋅ ⋅

6. (a)499∑

k=0

1

2(k + 1)

(b)∞∑

k=0

(−1)k

2k

(c)∞∑

k=0

(−x)k

k!

(d)∞∑

k=0

x2k+1

(2k)!

7. (a) 1 + 4x+ 6x2 + 4x3 + x4

(b) 1− 32y + 3

4y2 − 1

8y3

(c) 81x4 − 216x3y + 216x2y2 − 96xy3 + 16y4

8. (a)√1− x = 1− 1

2x− 1

8x2 − 1

16x3 + ⋅ ⋅ ⋅ valid for −1 < x < 1

(b)1

1 + 2x= 1− 2x+ 4x2 − 8x3 + ⋅ ⋅ ⋅ valid for ∣x∣ < 0.5

9. (a) e−2x = 1− 2x+ 2x2 − 43x3 + ⋅ ⋅ ⋅ valid for all x

(b) cos(3x) = 1− 92x2 + 27

8x4 − 81

80x6 + ⋅ ⋅ ⋅ valid for all x

(c) ln(1− 2x) = −2x− 2x2 − 83x3 − 4x4 + ⋅ ⋅ ⋅ valid for −0.5 ≤ x < 0.5

(d) ex2= 1 + x2 + 1

2x4 + 1

6x6 + ⋅ ⋅ ⋅ valid for all x

(e) sin(x/4) = 14x− 1

384x3 + 1

122880x5 − 1

82575360x7 + ⋅ ⋅ ⋅ valid for all x

10. Take x = 0.12, then√0.88 ≈ 1− 1

2(0.12)− 1

8(0.12)2− 1

16(0.12)3 = 0.938092. The error

is√0.88− 0.938092 = −0.88480× 10−5

11. I ≈ E

Lt− ER

2L2t2 +

ER2

6L3t3, lim

R→0I =

E

Lt

12. x− 1− 12(x− 1)2 + 1

3(x− 1)3 − 1

4(x− 1)4, ln(1.1) ≈ 0.09530833 (8dp).

45

Engineering MathematicsUFMFJ9-30-1

Worksheet 4Solutions

H1: (iii) sin−1 x ≈ x+ 16x3 + 3

40x5 + 5

112x7

sin−1(0.5) ≈ 0.5 + 16(0.5)3 + 3

40(0.5)5 = 0.523177

(iv) sin−1(0.5) = 16¼. So we find ¼ ≈ 6× 0.523177 = 3.139062.

This is accurate to 2dp.

H2: Hint: f(x+ ℎ) = f(x) + ℎf ′(x) + 12!ℎ2f ′′(x) + 1

3!ℎ3f ′′′(x) +O(ℎ4).

H3: g(x) ≈ 56x3 − 17

120x5, lim

x→0

g(x)

x3= 5

6

46

Engineering Mathematics Worksheet 5 UFMFJ9-30-1 Solutions

Answers to Worksheet 5: Complex Numbers

47

Engineering Mathematics Worksheet 5 UFMFJ9-30-1 Solutions

48

Engineering Mathematics Worksheet 6: Integration

UFMFJ9-30-1 Solutions

1

C&D refers to Croft and Davison Mathematics for Engineers Third Edition.

1. (a) Cxxxx ++−− 32/53/4/234

(b) Cxx

++−2/3

62

6

1

(c) ( ) Cxex

++−−

2cos232

If you have problems with Qu 1 & Qu 2, then go on and try the exercises on p783 of C&D, plus the End

of Block exercises on p784.

2. (a) 2 (b) 0.318 (Hint: rearrange algebraically before integrating)

(c) 3.729 (d) 5

1 cos5

T n

n T

π

π

−

3. (a) 0.75

(b) 0.1698 = 8

15π (Hint: use ( ) ( ) ( ) ( )

1 1sin sin cos cos

2 2A B A B A B= − − + )

(c)

−

T

n

n

T

2sin

2

1 π

π (Hint: use ( ) ( )2 1 1

sin cos 22 2

x x= − )

If you have difficulty with Qu 8 then have a go at the exercises and End of Block exercises on C&D

p839.

4. (a) 0.637 = 2

π (b) 2.350 = 1e e−

(c) 10 3 (d) 0.5 (Hint: use ( ) ( )2 1 1cos cos 2

2 2x x= + )

If you have problems with Qu 3 & Qu 4 then go on and try the exercises on p789 of C&D, plus the End

of Block exercises on p791.

5. (a) r.m.s. value = 1 1=

(b) r.m.s. value = 8

0.730296715

=

(c) r.m.s. value = 1 2 1 2 0.70710678= =

(d) r.m.s. value = 1 2 1 2 0.70710678= =

If you had problems with Qu 2 then go on to try the exercise in C&D p888

49



2

6. (a) Integrate by parts: [ ]b b

b

a

a a

dv duu dx uv v dx

dx dx= −∫ ∫

Choose u x= and ( )cosdv

nxdx

= so 1du

dx= and ( )

1sinv nx

n=

Hence ( ) ( ) ( )22 2

00 0

1 1cos sin sinx nx dx x nx nx dx

n n

ππ π

= − ∫ ∫

( ) ( )2 2

2 2 2

0 0

1 1 1 1sin cos sin cos

2 2 2

n nx nx nx

n n n n n

π ππ π π

= + = + −

(b) Integrate by parts twice.

( ) ( )2

2

3 2 2

2

2 2cos sin cos cos

4 2 2

n nx nx dx n

n n n n

π

π

π π π π ππ

= − + −

∫

(c) Integrate by parts twice.

41 13

0.1904744 4

e−

= − =

7. (a) Substitute 2 4u t= + ; remember to change the limits.

( )2 8 8

0 4 4

3 3 1 3 1 32 1.03972

2 4 2 2 2dt du du n

t u u= × = = =

+∫ ∫ ∫ l .

(b) Substitute � = ��; remember to change the limits.

1)sin(2

1

2

1)sin()sin(

000

2

∫∫∫ ==×=

πππ

duuduudxxx .

(c) Substitute � = 2�� − 1; remember to change the limits.

0.451)15ln(6

11

6

1

6

11

12

15

1

15

1

2

1

3

2

===×=− ∫∫∫ du

udu

udt

t

t (3 dp).

(d) Substitute � = ln(�); remember to change the limits.

2

1)ln(1

01

== ∫∫ duudxx

xe

.

50



3

8. (a) Use partial fractions to re-write the integral.

( )( )

1 1 1 14 2 5

4 2 5 4 2 5 2

xdx dx n x n x c

x x x x

+ = − = + − + +

+ + + + ∫ ∫ l l

(b) Use partial fractions.

( )( )( ) ( ) ( ) ( )

21 1 16

4 4 1 1 9 1 45 4 1 25 4

xdx dx

x x x x x x

= − + + + + + + + +

∫ ∫

1 1 16

1 4 1 49 180 45

n x n x n x c= − + + + + + +l l l

(c) Use partial fractions.

( ) ( )

2

2 2

1 1 1 5

2 4 2 4 2

xdx dx

x x x x x

+= − + + + +

∫ ∫

1 1 5

24 2 4

n x n x cx

= − − + + +l l

(d) Use partial fractions.

( ) ( )

2

2 2

1 1 1 1

11 1

x xdx dx n x c

x xx x x

+ += − = + +

++ + ∫ ∫ l

9. (a) Volume = ( )22

22

00

12 6.283185

2t dt tπ π π

= = =

∫

(b) Volume = ( ) ( )( )2

0 0

1sin 1 cos 2

2t dt t dt

π π

π π= −∫ ∫

( )2

0

1sin 2 4.9348

2 2 2t t

ππ π

= − = =

10. (�, ��) = �� , 0�

11. ��

�

12. (a) ��

� (b) ��

�

51

UFMFJ9-30-1 Mathematical Formula Sheet

Trigonometry

Common trigonometric identities:

tan θ =sin θ

cos θ, sec θ =

1

cos θ, cosec θ =

1

sin θ, cot θ =

1

tan θ

cos2 θ + sin2 θ = 1, 1 + tan2 θ = sec2 θ, cot2 θ + 1 = cosec 2θ

sin(−θ) = − sin θ, cos(−θ) = + cos θ, tan(−θ) = − tan θ

sin(x± y) = sin x cos y ± cosx sin y

cos(x± y) = cosx cos y ∓ sin x sin y

tan(x± y) =tanx± tan y

1∓ tanx tan y

sin 2x = 2 sinx cos x

cos 2x = cos2 x− sin2 x = 2 cos2 x− 1 = 1− 2 sin2 x

tan 2x =2 tanx

1− tan2 x

Trigonometric angles of any magnitude

90o

270o

180o 0o or 360o

A ll positive

C osine positive

S ine positive

T angent positive

If R sin(ωt+ α) = a sinωt+ b cosωt then a = R cosα, b = R sinα, R =√a2 + b2

Graphs of common functions

Linear: y = mx+ c, m = gradient, c = vertical intercept

-

6

��

s

s

(x1, y1)

(x2, y2)

y

x

m =y2 − y1x2 − x1

The equation of a circle centre (a, b), radius r

-

6

��&%'$r r

(a, b)

(x, y)

y

x

r (x− a)2 + (y − b)2 = r2

52

The modulus function The unit step function, u(x)

|x| ={

x if x ≥ 0−x if x < 0

u(x) =

{1 if x ≥ 00 if x < 0

-

6

��

@@

@@y = xy = −x

y = |x|

x-

61

u(x)

x

Trigonometric functions

The sine and cosine functions areperiodic with period 2π.The tangent function is periodicwith period π.

Exponential functions

Graphs of y = ex and y = e−x showing exponential growth and exponential decay respectively.

Logarithmic functions

Graphs of y = ln x (dashed) and y = log10 x (solid).

53

Algebra

Quadratic equations

If ax2 + bx+ c = 0 then x =−b±

√b2 − 4ac

2aprovided a 6= 0.

Completing the square:

If a 6= 0, ax2 + bx+ c = a

(x+

b

2a

)2

+4ac− b2

4a

Indices

xaxb = xa+b xa

xb= xa−b x−n =

1

xn(xa)b = xab x

1n = n

√x x0 = 1

Logarithms

log a + log b = log ab log a− log b = log(a/b) log an = n log a

If x = an then n =ln x

ln awhere ln x denotes the logarithm of x to base e.

Partial fractions

Assuming the numerator f(x) is of a less degree than the relevant denominator, constants a, b and care distinct and q2 − 4pr < 0, the following identities are typical examples of the form of partialfractions used:

f(x)

(x+ a)(x+ b)≡ A

(x+ a)+

B

(x+ b)

f(x)

(x+ a)(x+ b)(x+ c)≡ A

(x+ a)+

B

(x+ b)+

C

(x+ c)

f(x)

(x+ a)2(x+ b)≡ A

(x+ a)+

B

(x+ a)2+

C

(x+ b)

f(x)

(px2 + qx+ r)(x+ a)≡ Ax+B

(px2 + qx+ r)+

C

(x+ a)

Numerical methods for approximating areas

In order to approximate the area A =∫ b

ay dx, sub-

divide the interval [a, b] into N equally spaced in-tervals, each of length h. Denote xi = a+ ih, yi =y(xi) for i = 0, 1, 2, . . . , N , where h = (b− a)/N .

-

6y

x

a b

x0 x1 x2 xN

y0 y1 y2 yN

-�h

Trapezoidal rule:

Area =h

2

[y0 + 2(y1 + y2 + · · · + yN−1) + yN

]

Simpson’s rule: (Note: N must be even)

Area =h

3

[y0 + 4(y1 + y3 + · · ·+ yN−1) + 2(y2 + y4 + · · ·+ yN−2) + yN

]

54

Sequences and Series

Arithmetic progression: a, a+ d, a+ 2d, . . .a = first term, d = common difference, kth term = a+ (k − 1)dSum of n terms, Sn = n

2(2a + (n− 1)d)

Geometric progression: a, ar, ar2, . . .a = first term, r = common ratio, kth term = ark−1

Sum of n terms, Sn =a(1− rn)

1− r, provided r 6= 1

Sum of an infinite geometric progression:

S∞ =a

1− r, provided −1 < r < 1.

The binomial theorem

If n is a positive integer

(1 + x)n = 1 + nx+n(n− 1)

2!x2 +

n(n− 1)(n− 2)

3!x3 + · · ·+ xn

When n is negative or fractional, the series is infinite and converges when −1 < x < 1

MacLaurin’s series

f(x) = f(0) + xf ′(0) +x2

2!f ′′(0) +

x3

3!f ′′′(0) + · · ·

Taylor’s series

f(x) = f(a) + (x− a)f ′(a) +(x− a)2

2!f ′′(a) +

(x− a)3

3!f ′′′(a) + · · ·

Exponential series

ex = 1 + x+x2

2!+

x3

3!+ · · · (valid for all values of x)

Logarithmic series

ln(1 + x) = x− x2

2+

x3

3− x4

4+ · · · (valid if − 1 < x ≤ 1)

Trigonometrical series

sin x = x− x3

3!+

x5

5!− x7


cos x = 1− x2

2!+

x4

4!− x6


Complex numbers

-

6

��

r

θ

s

xa

y

b (a, b)zz = a+ jb = r(cos θ + j sin θ)

= r∠θ = rejθ where j2 = −1

Modulus: r = |z| =√a2 + b2

Argument: θ = arg z is such that tan θ = b/a

Addition: (a+ jb) + (c+ jd) = (a+ c) + j(b+ d)

Subtraction: (a+ jb)− (c+ jd) = (a− c) + j(b− d)

Complex equations: If m+ jn = p+ jq then m = p and n = q

Multiplication: z1z2 = r1r2∠(θ1 + θ2)

Division:z1z2

=r1r2∠(θ1 − θ2)

De Moivre’s theorem: (r∠θ)n = rn∠nθ= (r(cos θ + j sin θ))n = rn(cosnθ + j sin nθ)

55

Matrices and determinants

Matrices:

If A =

(a bc d

)and B =

(e fg h

)then

A+B =

(a+ e b+ fc+ g d+ h

), A− B =

(a− e b− fc− g d− h

)

AB =

(ae+ bg af + bhce+ dg cf + dh

), A−1 =

1

ad− bc

(d −b

−c a

)provided ad− bc 6= 0

Determinants:

Second order∣∣∣∣a bc d

∣∣∣∣ = ad− bc

Third order∣∣∣∣∣∣

a1 b1 c1a2 b2 c2a3 b3 c3

∣∣∣∣∣∣= a1

∣∣∣∣b2 c2b3 c3

∣∣∣∣− b1

∣∣∣∣a2 c2a3 c3

∣∣∣∣+ c1

∣∣∣∣a2 b2a3 b3

∣∣∣∣

Eigenvalues and Eigenvectors

Eigenvalues λ and eigenvectors x of matrix A satisfy Ax = λx.

To find the eigenvalues, solve |A− λI| = 0.

Vectors

PPPPPPPq

��1θ

a

b6e

If r = xi+ yj+ zk then |r| =√

x2 + y2 + z2

Let a = a1i+ a2j+ a3k and b = b1i + b2j+ b3k

Scalar or dot product:

a · b = a1b1 + a2b2 + a3b3

a · b = |a||b| cos θ

Vector or cross product:

a × b = |a||b| sin θewhere e is a unit vector perpendicular to the plane containing a and b in a sense defined by the righthand screw rule.

a× b =

∣∣∣∣∣∣

i j k

a1 a2 a3b1 b2 b3

∣∣∣∣∣∣= (a2b3 − a3b2)i+ (a3b1 − a1b3)j+ (a1b2 − a2b1)k

Vector equation of a line: r = a+ t(b− a)passing through points with position vectors a and b.

Vector equation of a plane: r · n = a · n

which contains the point with position vector a and being perpendicular to the vector n.

56

Differentiation

Standard derivatives:

y = f(x)dy

dxor f ′(x)

1 0

x 1

xn nxn−1

sin ax a cos ax

cos ax -a sin ax

tan ax a sec2 ax

eax aeax

ln(ax)1

x

Product rule:d

dx(uv) = u

dv

dx+ v

du

dx

Quotient rule:d

dx

(uv

)=

vdu

dx− u

dv

dxv2

Chain rule:dy

dx=

dy

du× du

dx

Implicit differentiation:d

dx[f(y)] =

d

dy[f(y)]× dy

dx

Parametric differentiation: If x = x(t) and y = y(t) then

dy

dx=

y

xand

d2y

dx2=

d

dt

(dy

dx

)

xwhere x =

dx

dtand y =

dy

dt

Maximum and minimum values:

If at a point x1,dy

dx= 0 and

d2y

dx2> 0 then it is a local minimum.

If at a point x2,dy

dx= 0 and

d2y

dx2< 0 then it is a local maximum.

If at a point x3,d2y

dx2= 0 and

dy

dx6= 0 then it is a point of inflexion.

57

Differential equations

1. If ad2y

dx2+ b

dy

dx+ cy = 0 where a, b, c are constants then

(i) Solve the auxiliary equation am2 + bm+ c = 0

(ii) If the roots of the auxiliary equation are:

(a) real and different, say m = α and m = β, then the general solution is

y = Aeαx +Beβx

(b) real and equal, say m = α, then the general solution is

y = (Ax+B)eαx

(c) complex, say m = α± jβ, then the general solution is

y = eαx(A cos βx+B sin βx)

(iii) Given boundary conditions, constants A and B can be determined and the particularsolution obtained.

2. If ad2y

dx2+ b

dy

dx+ cy = f(x) where a, b, c are constants then

(i) Solve the auxiliary equation am2 + bm+ c = 0

(ii) Obtain the complementary function (C.F.), ycf , as in 1(ii) above

(iii) To find the particular integral, ypi, first assume a particular integral which is suggested byf(x) but which contains undetermined coefficients. The table below gives some suggestedsubstitutions.

(iv) Substitute the suggested particular integral into the original differential equation and equaterelevant coefficients to find the constants introduced

(v) The general solution is given by

y = ycf + ypi

(vi) Given boundary conditions, constants A and B can be determined and the particularsolution obtained.

Form of the particular integral for different functions

f(x) Trial solution

K ypi = CK + Lx ypi = C +DxK + Lx+Mx2 ypi = C +Dx+ Ex2

Kepx ypi = Cepx

K sin px+ L cos px ypi = C sin px+D cos px

where f(x) is not part of the complementary function. If f(x) is part of the complementaryfunction, try x times the usual trial solution as the particular integral.

58

Integration

Standard integrals:

y(x)

∫y dx

xnxn+1

n+ 1+ c (provided n 6= −1)

1

xln |x|+ c

cos ax1

asin ax+ c

sin ax −1

acos ax+ c

eaxeax

a+ c

1

ax+ b

1

aln |ax+ b|+ c

1

(ax+ b)2−1

a(ax+ b)−1 + c

Integration by parts: If u and v are both functions of x then:∫

udv

dxdx = uv −

∫vdu

dxdx

Integration by substitution:∫

f(u)du

dxdx =

∫f(u) du

Applications of integration:

1. Area under a curve, A =

∫ b

a

y dx

-

6y

xx = a x = b

A

y = f(x)

��

��

��

��

��

��

��

��

��

��

��

��

��

��

2. The mean value of y = f(x) between x = a and x = b is given by:

mean value =1

b− a

∫ b

a

y dx

59

3. The root mean square value of y = f(x) between x = a and x = b is given by:

r.m.s. value =

√(1

b− a

∫ b

a

y2 dx

)

4. Centre of Mass

This is the point in a body such that an external force produces an acceleration just as thoughthe whole mass were concentrated there. Let (x, y) be the co-ordinates of the centre of mass ofa system of particles each of mass mi, and centres of mass located at (xi.yi). Then

x =

∑mixi∑mi

, y =

∑miyi∑mi

5. Volume of Revolution

If the graph of y(x), between x = a and x = b is rotated about the x−axis, the volume of thesolid formed is

∫ b

a

πy2 dx

6. Moments of Inertia

A small element of mass δm is located r from the axis through the origin. The moment of inertiaof this element rotated perpendicular to the axis is r2 δm.

Given a lamina of mass m with m =∑

δm, the total moment of inertia about the axis is

∑r2 δm

As δm → 0, the moment of inertia, I, becomes

I =

∫r2 dm,

where the limits are chosen to include the entire lamina.

60

Laplace transforms

function transform

f(t) F (s) =

∫∞

0

e−stf(t) dt

f ′(t) sF (s)− f(0)

f ′′(t) s2F (s)− sf(0)− f ′(0)

f(t− a)H(t− a) e−asF (s)

f(t)e−at F (s+ a)

11

s

t1

s2

t22

s3

tnn!

sn+1

e−at1

s+ a

te−at1

(s+ a)2

tne−atn!

(s+ a)n+1

sin ata

s2 + a2

cos ats

s2 + a2

e−bt sin ata

(s+ b)2 + a2

e−bt cos ats+ b

(s+ b)2 + a2

∫ t

0

f(p)g(t− p)dp F (s)G(s)

Initial value theorem: lims→∞

sF (s) = f(0), lims→∞

s(sF (s)− f(0)) = f ′(0)

Final value theorem: lims→0

sF (s) = limt→∞

f(t), lims→0

s(sF (s)− f(0)) = limt→∞

f ′(t)

61

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