# Engineering Mathematics

**The textbook for Engineering Mathematics is:**

Complete solutions to all the problems in the book

**Topic Videos and Test Questions**

**All the test questions below were developed by Dr Martin Greenhow and his team at Brunel University London. (You can enter any name and any number like 123 for your student number. Also don’t need to give your email address, just enter a school where it asks for your email address.) You can do the test as many times as you wish as the numbers change every time you refresh the page. **

There are also test questions from Newcastle University using Numbas and these were developed by Bill Foster and Christian Perfect.

**Mock first assignment – engineering mathematics**

**Mock Test on second week of engineering mathematics**

**Engineering Mathematics Mock Exam**

#### Chapter Introduction Arithmetic for Engineers

**Test on Indices of Arithmetic - Section Intro B**

**Test on Arithmetic Indices - Section Introduction B**
**Test on Arithmetic of Fractions - Section Introduction D**
**Test on Arithmetic of Fractions - Section Introduction D**
**Test on Arithmetic of Fractions (Applied) - Section Introduction E**
**Test on Decimals - Section Introduction F**
**Test on Standard Form - Section Introduction G**
**Test on Arithmetic Operations - Introductory Chapter Section I**
**Test on Percentages - Section Introduction J**
**Test on Ratios - Section Introduction K**

#### Chapter 1 Engineering Formulae

**Test on Linear Equations**
**Test on Transposing Formulae - Section 1B
Test Questions on Transposing of Formulae - Section 1B
Test on Rearranging formulae - Section 1B and 5B
**
**Test on Simplification of Algebra of Fractions- Section 1B**
**Test on Algebraic Fractions - Section 1B**
**Test on Indices - Section 1C**
**Test on Algebraic Indices - Section 1C**
**Test on Dimensional Analysis - Section 1D**
**Test on Expanding Brackets - Section 1E
Test on Expanding Brackets - Section 1E
Test on Factorisation - Section 1F
Test on Quadratic Equations - Section 1G
Test on Solving Quadratic Equations - Section 1G
Test on Simultaneous Equations - Section 1H
**

#### Chapter 2 Visualizing Engineering Formulae

**Test on Completing the Square - Section 2D
Test on Completing the Square - Section 2D
Test on the binomial expansion using Pascal's triangle - Section 2F**

#### Chapter 3 Functions in Engineering

Chapter 3: Functions Page 137-38

Chapter 3 Functions Pages 138-140

**Test on Evaluation of Functions Section 3A**

Chapter 3 Inverse functions pages 141 – 44

**Test on Inverse Functions Section 3b**

**Test on Composite Functions Section 3d**

Chapter 3: Limits of Functions pages 159-64

Test on Limits of functions Section 3E

T**est on Limits of Functions – Section 3E**

**Test on Limits of Functions II – Section 3E**

###### Chapter 4 Trigonometry and Waveforms

Chapter 4A Trigonometric Functions Pages 172-76

Chapter 4A Inverse Trigonometric Functions Pages 177-79

**Test on Trigonometric Functions – Section 4A**

**Test on Trigonometric Ratios – Section 4A**

Chapter 4B Angles and Graphs pages 181-186

Chapter 4E Radians pages 201-203

**Test on Radians - Section 4E **

Chapter 4G Trigonometric Identities pages 21-218

Exercise 4g questions 3(i), 4 and 6(i) on page 218-19

Chapter 4H Applications of Identities pages 219-222

Section 4h Engineering examples of trig identities

Chapter 4I Conversion pages 223-229

Section 4i Conversion Examples

Sections 4h and 4i examples.pdf

Exercise 4(i) question 1(iv) and 2(i) page 230

Exam questions 4 questions 3, 10 and Miscellaneous 4 question 5

Exam questions 4 and miscellaneous exercise 4 notes

**Test on Conversion of Trigonometric Functions - Section 4I**

#### Chapter 5 Logarithmic, Exponential and Hyperbolic Functions

Chapter 5B The Exponential Function pages 239-240

Chapter 5B Exponential Function pages 240-244

Section 5B Exponential Function examples with subtitles

Section 5B Exponential function examples part II with subtitles

Section 5B Exponential function examples part II with subtitles

**Test on Exponential Functions**

Chapter 5C The Logarithmic Function pages 245-248

Chapter 5C Logarithmic Function pages 247-252

Section 5C Logarithmic Function Examples with subtitles

Section 5C Logarithmic Function Examples with subtitles

Video content

- 00:06 Exam question
- unknown on ceiling

Section 5C Logarithmic Function Examples with subtitles

Video content

- 00:56 log button on caculator
- 02:25 example
- 05:04 other bases
- 06:00 financial example

**Test on Logarithmic Function - Section 5C**

Chapter 5E Hyperbolic Functions pages 259-261

Chapter 5E Hyperbolic Functions pages 262-266

Hyperbolic Functions I Examples with subtitles

Hyperbolic Function Examples II with subtitles

Video content

- 01:23 Exam questions 5 question 13
**b** - 08:00 Engineering question 13 of Exercise 5(e)

**Test on Hyperbolic Functions - Section 5E**

#### Chapter 6 Differentiation

Chapter 6A The Derivative pages 272-274

Chapter 6A The Derivative pages 274-279

Chapter 6B Derivatives of Functions Pages 282-85

Chapter 6B Derivatives of Functions pages 286-89

**Differentiating Algebraic Functions - Section 6B
Test on the chain rule section 6B
Test on Chain rule of functions - Section 6B
Test on Chain Rule - Section 6B
**

Chapter 6C Chain Rule Revisited pages 290-95 with subtitles

Video content

- 03:00 explanation why this method is easier
- 03:57 Exercise 6c question 1(c)
- 05:20 Exercise 6c question 1(e)
- 07:45 Exam question 6 question 4(iv)
- 10:40 Miscellaneous Exercise 6 question 8

Section 6C Engineering Examples using the chain rule with subtitles

Video content

- 00:28 Miscellaneous Exercise 6 question 12
- 04:13 Miscellaneous Exercise 6 question 13
- 09:36 The derivative of sin(ωt)

##### Chapter 6D Product and Quotient Rules pages 296-302

Section 6D Product and Quotient Rule examples with subtitles

Video content

- 01:13 Derivative of
*e*sin(2∑)^{∑} - 05:35 Examination questions 6 question 4
**ii**

Section 6D Product and Quotient Rule examples II with subtitles

Video content

- 00:47 Quotient rule formula
- 01:55 Exercise
**6(d)**question 3**k**

Section 6D: Product and Quotient Rule Exam Questions with subtitles

Video content

- 00:22 Differentiating 5
*x/*(1+*e*)^{x} - 04:57 Examination questions 6 question 5
**i**

**Product and Quotient rule examples.pdf**

**Test on Product Rule - Section 6D**
**A more challenging test on the product rule - section 6D**
**Test on Quotient Rule - Section 6D**
**More questions on the quotient rule - section 6D**
**A more challenging test on the quotient rule - section 6D**
**Test on Differentiating Exponential Functions - Section 6D**

Chapter 6E Higher Derivative pages 302-306

Section 6E Higher Derivatives Examples with subtitles

Video content

- 01:47 Exercise 6(e) question 6
- 08:47 Examination questions 6 question 8

Chapter 6F Parametric Differentiation pages 308-314

Section 6F Parametric Differentiation Examples with subtitles

Video content

- 01:40 Exercise 6(f) question 2
- 03:50 d
*y*/d*x*formula for parametric equations - 08:24 Exercise 6(f) question 9
- 11:38 d
^{2}*y*/d*x*^{2}

**Test on Parametric Differentiation - Section 6F**

Section 6G Implicit Differentiation Page 315-318

Section 6G Implicit and log differentiation examples with subtitles

Video content

- 01:13 Exercise
**6(g)**question**2c** - 06:32 Idea of logs
- 06:52 Napier
- 07:34 Exercise
**6(g)**question**5**a

One question on Implicit Differentiation – Section G1 of chapter 6

**Test on Implicit Differentiation - Section 6G**

Section 6g Logarithmic Differentiation pages 319-321

Logarithmic Differentiation Notes

Test on Logarithmic Differentiation – Section 6G

**Test on Differentiation - Chapter 6**

#### Chapter 7 Engineering Applications of Differentiation

Section 7A Curve Sketching pages 327-334

Section 7A Curve Sketching Notes

Section 7a Exam questions 7 page 394 with subtitles

Video content

- 01:02 Curve Sketching
- 03:40 Examination questions
**7**question**2a** - 11:05 Crosses the axes

Exercises 7a question 10 with subtitles

Video content

- 00:11 Exercise
**7(a)**question**10** - 06:13 Point of inflexion
- 13:00 Sketch of graph

Section 7B Optimization pages 339-342

Section 7B Exercises 7b question 4

Section 7C First Derivative Test pages 345-348

Section 7c Exercises 7c question 2 page 348

Section 7F Series Expansion pages 358 to 362

Notes on Section 7F Series Expansion pages 358 to 362

Section 7f Series Expansion pages 362 to 363

Notes on Section 7f Series Expansion pages 362 to 363

F2: Maclaurin series of functions pages 364-65

Section 7F: Maclaurin Series for sin(x) Pages 363-365

Section F3: Taylor series of functions pages 366-369

Section G: Binomial Revisited pages 369-70

Section 7G: Binomial Revisited Pages 369-73

**Test on Binomial Series - Section 7G**

Section 7H: Introduction to Infinite Series Pages 375-77

Section 7H: Introduction to Infinite Series pages 377-79

Section 7H3: Examples of Geometric Series pages 379-83

**Test on Geometric Series - Section 7H3**

Section 7H4: Ratio Test pages 383-86

Section J Power Series – Ratio Test

Section 7J: General Power Series

Section 7J Interval of Convergence of Power Series

Section 7J Interval of Convergence examples

#### Chapter 8 Integration

Section 8A and 8B pages 400-413

Sections 8A and 8B Notes

**Test on basic indefinite integration - Section 8A**

The notes for 8b are here: SECTION 8B.pdf

Section 8C Part I pages 414-417

Test on Integration by Substitution – Section 8C

Section 8C Part II An Important Integral pages 417-420

The notes for 8c are here: SECTION 8C.pdf

**Test on integration by substitution with algebraic integrand- Section 8C**

Section 8D Engineering Applications of Integration pages 421 -426

The notes for 8d are here: SECTION 8D.one

The notes for 8d are here: SECTION 8D.pdf

The notes for exam questions is here EXAM QUESTIONS 8.pdf

Section 8E Integration by Parts pages 432-438

Integration by parts, Exercises 8e page 440

Integration by parts Exam questions page 466

Here are the notes for the above videos: section 8e.pdf

**Test on Integration by Parts - Section 8E
****Six questions on integration by parts - section 8E**
**Test on integration by parts with exponential function - Section 8E**
**Test on integration by parts involving logarithmic function - Section 8**E
**Test on challenging questions on integration by parts - Section 8E**
**Test on integration by parts with trigonometric functions- Section 8E**

Section 8F Algebraic Fractions pages 440-446

Section 8F Improper Fractions pages 446-449

Integration by partial fractions Exercises 8g page 455

Exam questions on integration by partial fractions pages 466-67

Here are the notes for the above videos: Section 8g.pdf

**Test on Partial Fractions with Linear Factors - Section 8F
Test on Algebraic Fractions - Section 8F
**
**Test on Linear Partial Fractions - Section 8F**
**Test on Repeated Partial Fractions - Section 8F**
**Test on long divsion - Section 8F**

Section 8G: Integration by Partial Fractions pages 450-452

**Testing integration by partial fractions – section 8g**

Integration by Substitution Exercise 8h page 459

Here are the notes for the above videos: Section 8h.pdf

**Test on Integration using Partial Fractions - Section 8G**
**Test on Calculus - Chapters 6 and 8
**

Trigonometric Techniques for integration Exercise 8i page 459

Here are the notes for the above videos: section 8 I.pdf

Eight questions on integration by substitution – sections 8C, 8H and 8I

#### Chapter 9: Applications of Integration

Section 9C Engineering Applications of integration pages 489-492

Section 9C2 Mean and root mean square (RMS) values pages 492-95

Section 9C2 RMS values pages 495-97

Here are the notes for Section 9C section 9c.pdf

Exercise 9C Engineering applications of integration page 498

Here are the notes for Exercise 9C Section 9c examples.pdf

**Test on the Trapezuim Rule – Section 9A**

Section E3: OtherApplications of Integration pages 506-507

#### Chapter 10: Complex Numbers

Section 10a Arithmetic of complex numbers pages 514-523 Video

**Test on Arithmetic of Complex Numbers - Section 10A**

Section 10b Representation of complex numbers pages 525-531 Video

`T`**est on Modulus and Argument of a Complex Number - Section 10B**
Test on modulus, argument and conjugate of a complex number - Section 10B

Section 10c Multiplication and division in polar form pages 532-537 Video

**Test on conversion between polar and Cartesian - Section 10B and 10C**

Section 10D Powers and Roots of Complex numbers pages 538-543

Notes on Section 10D Powers and Roots of Complex numbers pages 538-543

Section 10D Powers and Roots of Complex Numbers Pages 544-547

Notes on Section 10D Powers and Roots of Complex Numbers Pages 544-547

Section 10E Exponential Form of Complex Numbers Pages 548-553

Notes on Section 10E Exponential Form of Complex Numbers Pages 548-553

**Test on Exponential Form of complex number - Section 10E**

#### Chapter 11 Matrices

**Test on Arithmetic of Matrices - Section 11A
Test on Manipulation of Matrices - Section 11A
Test on Manipulation of Matrices II - Section 11A
Test on Square Matrices - Section 11B and 11C
**
**Test on Gaussian Elimination - Section 11D**
**Test on Eigenvalues and Eigenvectors - Section 11F**

#### Chapter 13: First Order Differential Equations

**Test on Separable first-order differential equations - Section 13A**
The following topics are not covered in the textbook.
**Test on Bernoull Differential Equation**
**Test on Exact Differential Equation
Test on Converting First Order to Homogeneous Form
Test on Linear First Order Differential Equations - Sections 13A and 13B
Test on Ist Order Differential Equations -Section 13A and 13B
**Test on Numerical Methods on solving First Order ODEs

#### Chapter 14: Second Order Linear Differential Equations

Section 14a Homogeneous Differential Equations pages 732-737

Notes on Section 14a Homogeneous Differential Equations pages 732-737

*Online Test on homogeneous differential equations*
Test on Direct Integration applied to 2nd Order ODEs

Section 14b Applications pages 738-740

Notes on Section 14b Applications pages 738-740

Section 14b Applications pages 740 – 744

Notes on Section 14b Applications pages 740 – 744

**Test On Section 14B**

Section 14c Non- homogenous Differential Eqns pages 746-751

Notes on Section 14c Non- homogenous Differential Eqns pages 746-751

Section 14c Non-Homogeneous DES pages 752 to 757

Notes on Section 14c Non- homogenous Differential Eqns pages 752-57

**Test on Inhomogeneous equations Section 14C**

Section 14D Particular solutions to 2nd order DES pages 759 to 762

Notes on Section 14D Particular solutions to 2nd order DES pages 759 to 762

Section 14D Particular solutions to 2nd Order DES pages 762-63

Notes on Section 14D Particular solutions to 2nd Order DES pages 762-63

**Test on 2nd Order Differential Equations -Section 14D**

#### Chapter 16 Probability and Statistics

**Test on Data Summaries – Section 16B **

**Test on Probability Rules – Section 16C **

**Test on Probability Rules II – Section 16C **

**Test on Probability Rules III – Section 16C **

**Test on Combinations – Section 16D **

**T****est on Permutations – Section 16D **

**Additional material which is freely available at:**

Companion site for engineering mathematics

**A glossary is available at the following URL:**

**Matlab notes available at the following URL:**

Sample Engineering Mathematics Examination Questions:

Sample Examination Engineering Mathematics Questions

The following questions were developed by Laurence Taylor.

Here is a revision checklist was written by Laurence Taylor

Revision Checklist for Engineering Mathematics

** The following are not in the book.**

#### Chapter 17: Fourier Series

Some links on Fourier Series

A good link to visualize Fourier series is the following: Graphs

A useful application is here Sign your name in the circle and see how it is reproduced. If you click the theory link then you will see the relevance of Fourier series.

Another graphical link to Fourier series is the blog

Youtube video on Fourier series is here

###### Chapter 18 Laplace Transform

**Test on Applying Laplace Transforms to Differential Equations**

**Test on Inverse Laplace Transforms **

**Test on Properties of Laplace Transforms **

The year 2017-18 Videos

**Lecture 5 on Logarithmic Function Section 5C**

**Lecture 6 on Hyperbolic Functions Section 5E**

Lecture 7 Hyperbolic Functions Section 5E

### Chapter 6 Differentiation

** Derivatives of functions – Section 6B**

** Chain Rule Revisited – Section 6C**

**Product and Quotient Rules – Section 6D**

Section 6F Parametric Differentiation

Section 6F Parametric Differentiation Part II

Section 6f Part II Notes

#### A book on Differential Calculus by Jaromir Kuben

A good book for lots of examples in

#### Differential Calculus for Functions of a Single Variable by Jaromir Kuben

#### A book on Integral Calculus by Jaromir Kuben

This is an excellent interactive book:

**Integral Calculus**

Differential Calculus for Functions of a Single Variable by Jaromir Kuben