Let the fantastic wealth of resources below teach you all about Further Calculus.

Covers the whole of the Higher Maths course

Created by an experienced maths teacher

Resources used with students in Scottish Secondary Schools

Welcome to **highermathematics.co.uk**

A sound understanding of * Further Calculus* is essential to ensure exam success.

To access a wealth of additional ** free resources by topic** please either use the above

Passing the fast paced Higher Maths course significantly increases your career opportunities by helping you gain a place on a college/university course, apprenticeship or even landing a job. A ‘good’ pass at Higher Maths will set you up well for the AH Maths Course next year should you be interested. Please do your very best to keep on top of your studies.

For students looking for extra help with the Higher Maths course in 2020/21 you may wish to consider subscribing to the fantastic additional * exam focused* resources available in the Online Study Pack.

We hope the resources on this website prove useful and wish you the very best of success with your Higher Maths course in 2020/21.

Please find below:

2. Further Calculus – Worksheets

3. Further Calculus – Videos, Theory Guides & Mind Maps

4. Higher Maths Essential Skills

5. Higher Maths Exam Worksheets by Topic

6. Higher Maths Past & Practice Papers by Topic

7. Higher Maths Videos, Theory Guides, Mind Maps & Worksheets

8. Higher Maths Past & Practice Papers

9. 40 Non-Calculator Higher Maths Questions & Answers

10. 200 Higher Maths Questions & Answers

11. Practice Exam Papers A to H – Answers Included

12. 264 SQA Exam Multiple Choice Questions & Answers

13. Higher Maths Exam Check Lists

14. Old Higher Maths Exam Questions by Topic

15. Higher Maths Text Book Solutions

16. Higher Maths Theory Guides

18. Higher Maths Practice Unit Assessments – Solutions Included

19. Higher Maths Past Paper Video Solutions

20. Higher Maths Recommended Text Book

21. Exam Focused Online Study Pack – *For students looking for a ‘good’ Pass*

Rules for differentiating and integrating sinax & cosax are shown below

- Further Calculus at Higher Maths involves differentiating and integrating sinax & cosax
- The above rules are given in the Higher Maths Exam Formulae List
- If asked to differentiate sin³x ⇒ change to (sin x)³ and use the Chain Rule
- If asked to differentiate cos³x ⇒ change to (cos x)³ and use the Chain Rule

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The Chain Rule is used in Further Calculus

- The n goes to the front of the bracket
- Inside the bracket stays the same ⇒ ax + b
- Multiply by the differentiated first bracket ⇒ a
- Simplify by bringing the ‘a’ to the front of the bracket

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**Example 1**

- f (x) = (3x³ – 2x² + 5)³
- f ‘ (x) = 3 (3x³ – 2x² + 5)² (9x² – 4x)

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**Example 2**

- f (x) = sin³x = (sinx)³
- f ‘ (x) = 3(sinx)²(cos x) = 3sin²xcosx

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In Examples 1 and 2 above

- Power goes to the front
- Power is then deducted by 1
- First bracket stays the same
- Second bracket is the first bracket differentiated

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Further Integration is also included in Further Calculus

- Power is incremented up by 1 ⇒ to give ‘n + 1’
- Incremented power goes on the denominator ⇒ (n + 1)
- Incremented denominator power is multiplied by the x coefficient ⇒ to give ‘a(n + 1)’

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In the above

- Power is incremented up by 1 ⇒ to give 5
- Incremented power goes on the denominator ⇒ to give 5
- Incremented power on the denominator is multiplied by the x coefficient ⇒ 5 x 2 = 10

**2. Further Calculus – Worksheets**

Thanks to the SQA and authors for making the excellent resources below freely available. Please use regularly for revision prior to assessments, tests and the final exam.

**Differentiation – Worksheets**

Clear, easy to follow, step-by-step worked solutions to all worksheets below are available in the Online Study Pack.

Worksheets ____________________ | Topic _____________________________ | Answers ________ |

Essential Skills 4 | Equation of a Tangent to a Curve | Answers |

Essential Skills 5 | Stationary Points | Answers |

Essential Skills 17 | Derived Graphs | Answers |

Essential Skills 22 | Further Differentiation (Chain Rule) | Answers |

Higher Exam Questions | Derived Graphs | Answers |

Higher Exam Questions | Differentiation - 1 | Answers |

Higher Exam Questions | Differentiation - 2 | Answers |

Higher Exam Questions | Differentiation (Optimisation) | Answers |

Higher Exam Questions | Further Calculus | Answers |

**Integration – Worksheets**

Clear, easy to follow, step-by-step worked solutions to all worksheets below are available in the Online Study Pack.

Worksheets ____________________ | Topic __________________________ | Answers ________ |

Essential Skills 23 | Further Integration | Answers |

Essential Skills 25 | Differential Equations | Answers |

Essential Skills 26 | Definite Integrals | Answers |

Higher Exam Questions | Integration - 1 | Answers |

Higher Exam Questions | Integration - 2 | Answers |

Higher Exam Questions | Integration - Area under a Curve | Answers |

**3. Further Calculus – Videos, Theory Guides & Mind Maps**

Thanks to the authors for making the excellent resources below freely available. Please use regularly for revision prior to assessments, tests and the final exam.

**Differentiation**

Larbert Maths Videos __________________________________ | maths180.com Videos ______________________________________ | HSN Theory Guides ____________________________ | Mind Maps ______________________ |

Applications of Derivatives | Chain Rule - Increasing/Decreasing | Differentiation Theory (HSN) | Differentiation (HSN) |

Chain Rule | Optimisation - More Challenging Examples | Further Calculus Theory (HSN) | Preparing to Differentiate |

Closed Intervals | Rate of Change & Tangent to a Curve | Differentiation 1 | |

Complex Differentiation | Trig Functions & Chain Rule | Differentiation 2 | |

Curve Sketching | Further Differentiation | ||

Derived Graphs | |||

Equations of Tangents | |||

Fractions & Roots | |||

Increasing & Decreasing Functions | |||

Introduction to Differentiation | |||

Leibniz Notation | |||

Optimisation | |||

Stationary Points | |||

Sin x & Cos x |

**Integration **

Larbert Maths Videos __________________________ | maths180.com Videos ___________________________ | HSN Theory Guides __________________________ | Mind Maps ___________________ |

Integration - Introduction | Indefinite & Definite Integrals | Integration Theory (HSN) | Integration (HSN) |

Area Under a Curve | Area Under and Between Curves | Further Calculus Theory (HSN) | Further Calculus (HSN) |

Area Between Curves | Preparing to Integrate | ||

Differential Equations | Integration 1 | ||

Definite Integrals | Integration 2 | ||

Integrating Sin & Cos | |||

Integrating With The Chain Rule | |||

Integration - Further Examples | |||

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**4. Higher Maths Essential Skills **

Thanks to Mr G Rennie for making the excellent resources below freely available. The Essential Skills Worksheets can be used for general revision, homework, consolidation of a topic or preparation for assessments, tests and exams.

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**5. Higher Maths Exam Worksheets by Topic**

Thanks to the SQA and authors for making the excellent resources below freely available. The worksheets by topic are a fantastic study resource since they are actual past paper exam questions.

Clear, easy to follow, step-by-step worked solutions to all new CfE Higher Maths Questions below are available in the Online Study Pack.

**6. Higher Maths Past & Practice Papers by Topic**

Thanks to the SQA for making the excellent resources below freely available. Questions and answers have been split up by topic for your ease of reference.

Clear, easy to follow, step-by-step worked solutions to all questions below are available in the Online Study Pack.

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**7. Higher Maths Videos, Theory Guides, Mind Maps & Worksheets**

Dozens of Higher Maths Videos provide quality lessons by topic. Also included are excellent Theory Guides, Mind Maps and Revision Worksheets with actual Higher Maths exam questions.

Please click on our new *Higher Maths Videos & Worksheets* by topic dedicated page.

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**8. Higher Maths Past & Practice Papers**

Thanks to the SQA for making the excellent resources below freely available.

Clear, easy to follow, step-by-step worked solutions to all CfE Higher Papers below are available in the Online Study Pack.

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**9. 40 Non-Calculator Higher Maths Questions & Answers**

Thanks to the SQA and authors for making the excellent resources below freely available. Start with these questions to help build your confidence. Once finished, you may like to move onto the *200 Higher Maths Exam Questions* in the next section checking your answers as you go. If stuck, always ask your teacher for help as soon as possible.

Clear, easy to follow, step-by-step worked solutions to all 40 questions below are available in the Online Study Pack.

Exam Questions ______________________________________ | Answers ___________________________ |

Whole Booklet for Printing | Answers |

Sheet A - 10 Questions | Answers |

Sheet B - 10 Questions | Answers |

Sheet C - 10 Questions | Answers |

Sheet D - 10 Questions | Answers |

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**10. 200 Higher Maths Questions & Answers**

Thanks to the SQA and authors for making the excellent resources below freely available. Please try to do as many questions as possible, checking your answers as you go. If stuck, always ask your teacher for help as soon as possible.

Clear, easy to follow, step-by-step worked solutions to all 200 questions below are available in the Online Study Pack.

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**11. Practice Exam Papers A to H – Answers Included**

Thanks to the SQA and Larkhall Academy for making the excellent resources below freely available. Please use regularly for revision prior to assessments, tests and the final exam.

Clear, easy to follow, step-by-step worked solutions to Practice Papers A to E are available in the Online Study Pack.

Practice Paper _____________ | Paper 1 __________ | Paper 2 __________ | Papers 1 & 2 ____________ | Answers ___________ | |

Paper A | Paper 1 | Paper 2 | Papers 1 & 2 | HERE | |

Paper B | Paper 1 | Paper 2 | Papers 1 & 2 | HERE | |

Paper C | Paper 1 | Paper 2 | Papers 1 & 2 | HERE | |

Paper D | Paper 1 | Paper 2 | Papers 1 & 2 | HERE | |

Paper E | Paper 1 | Paper 2 | Papers 1 & 2 | HERE | |

Paper F | Paper 1 | Paper 2 | Papers 1 & 2 | HERE | |

Paper G | Paper 1 | Paper 2 | Papers 1 & 2 | HERE | |

Paper H | Paper 1 | Paper 2 | Papers 1 & 2 | HERE |

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**12. 264 SQA Exam Multiple Choice Questions & Answers **

Thanks to the SQA and authors for making the excellent resources below freely available. Multiple choice are primarily C level questions and a great place to start your revision. If stuck, always ask your teacher for help as soon as possible.

Questions __________ | Year ______ | Exam Multiple Choice __________________ | Exam Multiple Choice __________________ | Exam Multiple Choice __________________ |

1 - 20 | 2015 | Questions Only | Questions & Answers | Answers Only |

21 - 40 | 2014 | Questions Only | Questions & Answers | Answers Only |

41 - 60 | 2013 | Questions Only | Questions & Answers | Answers Only |

61 - 80 | 2012 | Questions Only | Questions & Answers | Answers Only |

81 - 100 | 2011 | Questions Only | Questions & Answers | Answers Only |

101 - 120 | 2010 | Questions Only | Questions & Answers | Answers Only |

121 - 264 | Mixed | Questions Only | Questions & Answers | Answers Only |

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**13. Higher Maths Exam Check Lists**

Thanks to the SQA and authors for making the excellent resources below freely available. These are fantastic check lists to assess your Higher Maths knowledge. Please try to use these regularly for revision prior to tests, prelims and the final exam.

Description ___________________________________________________ | Link ________ | Acknowledgements _________________________ |

Higher Maths Course Description | HERE | |

SQA Higher Maths Exam Formulae List | HERE | Courtesy of SQA |

Check List 1 - Higher Maths Exam Whole Course | HERE | Courtesy of Zeta Maths |

Check List 2 - Higher Maths Formulae List Not Given in Exam | HERE | |

Check List 3 - Higher Maths Exam Whole Course | HERE | |

Check List 4 - Higher Maths Exam Trig Revision Sheet | HERE | |

Check List 5 - One Page Summary Guide Unit 1 (HSN) | HERE | Courtesy of HSN |

Check List 6 - One Page Summary Guide Unit 2 (HSN) | HERE | Courtesy of HSN |

Check List 7 - One Page Summary Guide Unit 3 (HSN) | HERE | Courtesy of HSN |

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**14. Old Higher Maths Exam Questions by Topic**

Thanks to the SQA for making the excellent resources below freely available. The worksheets by topic are a fantastic additional study resource.

Topic ________ | Topic Name ___________________________ | Link ________ | Notes ___________________ |

Topic 1 | Circles | HERE | Answers Included |

Topic 2 | Differentiation | HERE | Answers Included |

Topic 3 | Exponentials & Logarithms | HERE | Answers Included |

Topic 4 | Functions | HERE | Answers Included |

Topic 5 | Further Calculus | HERE | Answers Included |

Topic 6 | Graphs of Functions | HERE | Answers Included |

Topic 7 | Integration | HERE | Answers Included |

Topic 8 | Polynomials | HERE | Answers Included |

Topic 9 | Quadratics | HERE | Answers Included |

Topic 10 | Recurrence Relations | HERE | Answers Included |

Topic 11 | Straight Line | HERE | Answers Included |

Topic 12 | Trig Addition Formulae | HERE | Answers Included |

Topic 13 | Trig Graphs & Equations | HERE | Answers Included |

Topic 14 | Vectors | HERE | Answers Included |

Topic 15 | Wave Function | HERE | Answers Included |

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**15. Higher Maths Text Book Solutions **

Thanks to the AHS for providing the Heinemann Higher Maths text book solutions below. These will prove extremely useful in helping to progress your Higher Maths knowledge. Please note that there may be the odd arithmetic error.

Topic ______________ | Topic Name ____________________ | |||||||||||||||||

Topic 1 | Straight Line | 1A | 1B | 1D | 1E | 1F | 1G | 1I | 1K | 1M | 1N | 1O | 1O | |||||

Topic 2 | Functions & Graphs 1 | 2A | 2B | 2C | 2D | 2F | 2G | 2I | ||||||||||

Topic 3 | Functions & Graphs 2 | 3A | 3C | 3K | 3N | 3O | 3P | |||||||||||

Topic 4 | Recurrence | 5A | 5B | 5B | 5C | 5D | 5F | 5H | 5I | |||||||||

Topic 5 | Differentiation | 6A | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 6J | 6L | 6M | 6N | 6O | 6P | 6Q | 6R | 6S |

Topic 6 | Polynomials | 7B | 7C | 7D | 7E | 7F | 7G | 7H | 7I | 7J | ||||||||

Topic 7 | Quadratics | 8C | 8D | 8E | 8F | 8H | 8I | 8J | 8K | |||||||||

Topic 8 | Integration | 9G | 9H | 9I | 9L | 9N | 9P | 9Q | ||||||||||

Topic 9 | Trigonometry | 11B | 11C | 11D | 11E | 11F | 11G | 11H | ||||||||||

Topic 10 | Circles | 12B | 12D | 12F | 12H | 12J | 12K | 12L | ||||||||||

Topic 11 | Vectors | 13A | 13C | 13D | 13E | 13F | 13G | 13I | 13K | 13L | 13M | 13N | 13O | 13P | 13Q | 13R | 13S | 13U |

Topic 12 | Further Calculus | 14B | 14C | 14C | 14E | 14G | 14H | 14I | 14J | 14K | ||||||||

Topic 13 | Exp's & Logs | 15C | 15D | 15E | 15F | 15G | 15H | 15I | 15I | 15J | 15K | 15L | ||||||

Topic 14 | Wave Function | 16A | 16C | 16D | 16E | 16F | 16G | 16H |

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**16. Higher Maths Theory Guides **

Thanks to HSN for making the excellent Higher Maths Theory Guides freely available for all to use. These will prove a fantastic resource in helping you consolidate your understanding of Higher Maths.

Theory Guides _________________ | Topic ____________________________________________ | Link _______ |

Theory Guide 1 | All Topics Unit 1 Theory (HSN) | HERE |

Theory Guide 2 | All Topics Unit 1 - One Page Summary Guide (HSN) | HERE |

Theory Guide 3 | All Topics Unit 2 Theory (HSN) | HERE |

Theory Guide 4 | All Topics Unit 2 - One Page Summary Guide (HSN) | HERE |

Theory Guide 5 | All Topics Unit 3 Theory (HSN) | HERE |

Theory Guide 6 | All Topics Unit 3 - One Page Summary Guide (HSN) | HERE |

Theory Guide 7 | All Topics Units 1,2 & 3 Theory (HSN) | HERE |

Theory Guide 8 | Circles Theory (HSN) | HERE |

Theory Guide 9 | Differentiation Theory (HSN) | HERE |

Theory Guide 10 | Exponentials & Logarithms Theory (HSN) | HERE |

Theory Guide 11 | Functions & Graphs Theory (HSN) | HERE |

Theory Guide 12 | Further Calculus Theory (HSN) | HERE |

Theory Guide 13 | Graphs Transformations Theory (Movement & Reflection) | HERE |

Theory Guide 14 | Graphs Transformations Summary Sheet | HERE |

Theory Guide 15 | Integration Theory (HSN) | HERE |

Theory Guide 16 | Polynomials & Quadratics Theory (HSN) | HERE |

Theory Guide 17 | Sequences Theory (HSN) | HERE |

Theory Guide 18 | Straight Line Theory (HSN) | HERE |

Theory Guide 19 | Trigonometry Theory (HSN) | HERE |

Theory Guide 19 | Vectors Theory (HSN) | HERE |

Theory Guide 20 | Wave Function Theory (HSN) | HERE |

**17. Higher Maths Mind Maps**

Thanks to the authors for providing the excellent resources below. These will prove a fantastic resource in helping you prepare for assessments, tests and the final exam.

Mind Map ____________ | Topic ____________________________ | Mind Map ___________ | Topic ____________________________ |

Mind Map 1 | Circle 1 | Mind Map 16 | Polynomials & Quadratics |

Mind Map 2 | Circle 2 | Mind Map 17 | Polynomials |

Mind Map 3 | Composite Functions | Mind Map 18 | Quadratics |

Mind Map 4 | Compound Angle Formula | Mind Map 19 | Recurrence Relations 1 |

Mind Map 5 | Differentiation (Preparation) | Mind Map 20 | Recurrence Relations 2 |

Mind Map 6 | Differentiation 1 | Mind Map 21 | Straight Lines 1 |

Mind Map 7 | Differentiation 2 | Mind Map 22 | Straight Lines 2 |

Mind Map 8 | Differentiation (Further) | Mind Map 23 | Trigonometry 1 |

Mind Map 9 | Functions & Graphs | Mind Map 24 | Trigonometry 2 |

Mind Map 10 | Functions | Mind Map 25 | Vectors 1 |

Mind Map 11 | Integration (Preparation) | Mind Map 26 | Vectors 2 |

Mind Map 12 | Integration 1 | Mind Map 27 | Vectors 3 |

Mind Map 13 | Integration 2 | Mind Map 28 | Wave Function 1 |

Mind Map 14 | Logs & Exponentials 1 | Mind Map 29 | Wave Function 2 |

Mind Map 15 | Logs & Exponentials 2 |

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**18. Higher Maths Practice Unit Assessments – Solutions Included**

Thanks to the authors for making the excellent resources below freely available for all to use. Please use regularly for revision prior to assessments, tests and the final exam.

Unit _________ | Paper ____________ | Acknowledgements _________________________ |

Unit One | Practice A | Courtesy of SPTA |

Unit One | Practice B | Courtesy of SPTA |

Unit Two | Practice A | Courtesy of SPTA |

Unit Two | Practice B | Courtesy of SPTA |

Unit Three | Practice A | Courtesy of SPTA |

Unit Three | Practice B | Courtesy of SPTA |

Unit Four | Practice A | Courtesy of SPTA |

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**19. Higher Maths Past Paper Video Solutions**

Please click DLB Maths to view Higher Maths Past Paper video solutions. This will prove an excellent resource in helping you prepare for assessments, tests and the final exam.

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**20. Higher Maths Recommended Text Book**

Please find below our highly recommended text book which can be ordered by clicking on the book/link.

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**21. Exam Focused Higher Maths Online Study Pack **

Through step-by-step worked solutions to exam questions available in the Online Study Pack we cover everything you need to know about how to calculate the * Further Calculus* to pass your final exam.

For students looking for a ‘good’ pass at Higher Maths in 2021 you may wish to consider subscribing to the fantastic additional * exam focused* resources available in the Online Study Pack. Subscribing to the Online Study Pack may therefore be one of your best ever investments. Please click Online Study Pack to view screenshots, examples and instructions on how to subscribe.

Please speak with your mum/dad and show them our fantastic website.

We hope the resources on this website prove useful and wish you the very best of success with your Higher Maths course in 2020/21.

Topic Links

Free resources to dozens of Higher Maths topics are available by clicking on any of the links to the right.