# Engineering Mathematics

Engineering Mathematics

The textbook for Engineering Mathematics is: http://www.palgrave.com/companion/Singh-Engineering-Mathematics-Through-Applications/  #### 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

Chapter 3: Limits of Functions pages 159-64

Test on Limits of functions Section 3E

Test 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 Ratios – Section 4A

Chapter 4B Angles and Graphs pages 181-186

Exercise 4e

`Test on Radians - Section 4E `

Chapter 4G Trigonometric Identities pages 21-218

Exercise 4g

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

Exercise 4e and 4g notes

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

Exercise 4i notes

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

Section 5B Exponential function examples part II

Section 5B Exponential function examples part II

`Test on Exponential Functions`

Chapter 5C The Logarithmic Function pages 245-248

Chapter 5C Logarithmic Function pages 247-252

Section 5C Logarithmic Function Examples

Section 5C Logarithmic Function Examples

Video content

• 00:06 Exam question
• unknown on ceiling

Section 5C Logarithmic Function Examples

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

Hyperbolic Functions I Examples

Hyperbolic Function Examples II

`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

Section 6B Chain Rule examples

Section 6B Chain rule exam questions

```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

```Section 6C Chain rule revisited examples
```
##### Chapter 6D Product and Quotient Rules pages 296-302
`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

Section 6e examples.pdf

Chapter 6F Parametric Differentiation pages 308-314

Section 6F Parametric Differentiation Examples

Section 6f.pdf

`Test on Parametric Differentiation - Section 6F`

Section 6G Implicit Differentiation Page 315-318

Section 6g Page 315 Notes

Section 6G Implicit and log differentiation examples

section 6g.pdf

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

Exercises 7a question 10

Section 7A Exam questions

section 7a.pdf

Section 7B Optimization pages 339-342

Section 7B Notes

Section 7B Exercises 7b question 4

Section 7b.pdf

Section 7C First Derivative Test pages 345-348

Section 7C Notes

Section 7c Exercises 7c question 2 page 348

Section 7c.pdf

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

Mac Series Notes

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

Section F3: Taylor series of functions pages 366-369

Taylor Series Notes

Section G: Binomial Revisited pages 369-70

Section 7G: Binomial Revisited Pages 369-73

Binomial Series notes

`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`

Exercises 8b page 413

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

Exercise 8c Page 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

Exercises 8d page 429

Exercises 8d part II page 429

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

Exercise 8d page 430

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

Exam questions 8 pages 466-67

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 8E

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

Five questions on chapter 8

#### 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

Improper Integrals

#### Chapter 10: Complex Numbers

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

section 10a notes

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

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

Section 10b notes

```Test 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

Section 10c notes

`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

`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

Additional material which is freely available at:

Companion site for engineering mathematics

A glossary is available at the following URL:

Glossary

Matlab notes available at the following URL:

Matlab

Sample Engineering Mathematics Examination Questions:

Sample Examination Engineering Mathematics Questions

Beer Barrel Problem

Old Formulae Sheets

The following questions were developed by Laurence Taylor.

Here is a revision checklist was written by Laurence Taylor

Revision Checklist for Engineering Mathematics

#### Chapter 17: Fourier Series

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

A useful application is 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

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

The Derivative – Section 6A

Section 6A lecture notes

Derivatives of functions – Section 6B

Section 6B notes

Chain Rule Revisited – Section 6C

Section 6C Notes

Product and Quotient Rules – Section 6D

Section 6D

Section 6E Higher Derivatives

Section 6E Notes

Section 6F Parametric Differentiation

Section 6F Notes

Section 6F Parametric Differentiation Part II

Notes

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

A good book for lots of examples in

#### 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