Let the fantastic wealth of resources below teach you all about the Nature of Turning Points.

Covers the whole of the AH Maths course

Created by an experienced maths teacher

Resources used with students in Scottish Secondary Schools

Welcome to **advancedhighermaths.co.uk**

A sound understanding of the * Nature of Turning Points* is essential to ensure exam success.

Study at Advanced Higher Maths level will provide excellent preparation for your studies when at university. Some universities may require you to gain a pass at AH Maths to be accepted onto the course of your choice. The AH Maths course is fast paced so please do your very best to keep on top of your studies.

For students looking for extra help with the AH 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 you find this website useful and wish you the very best with your studies in 2020/21.

Please find below:

1. AH Maths Past Exam Worksheets by Topic

4. Further Differentiation Resources

5. MIA Text Book Questions – Differentiation

6. MIA Text Book Questions** – **Further Differentiation

7. About the Nature of Turning Points

8. AH Maths Past and Practice Papers

9. AH Maths Past Paper Questions by Topic

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

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**1. AH Maths Past Paper Exam Worksheets by Topic**

Thanks to the SQA for making these available. The worksheets by topic below are an excellent study resource since they are actual SQA past paper exam questions.

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

**2. Links and Exam Formulae
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AH Maths - Home Page Link | HERE |

AH Maths - Whole Course Link | HERE |

AH Maths - Exam Formula List | HERE |

Higher Maths - Exam Formulae List | HERE |

AH Maths - Complete Formula List | HERE |

SQA Full Course Content and Outline | HERE |

**3. Differentiation Resources**

Thanks to the author(s) for making these available.

Differentiation Theory Guide 1 | HERE |

Differentiation Theory Guide 2 | HERE |

Differentiation HSN Summary Notes | HERE |

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**4. Further Differentiation Resources**

Thanks to the author(s) for making these available.

Further Differentiation Theory Guide 1 | HERE |

Further Differentiation Theory Guide 2 | HERE |

Further Differentiation HSN Summary Notes | HERE |

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**5. MIA Text Book Questions – Differentiation**

Recommended questions from the Maths In Action (2nd Edition) by Edward Mullan text book are shown below.

Worked solutions to all questions below are available in the Online Study Pack.

Subtopic ___________________________ | Page Number ____________ | Exercise ___________ | Recommended Questions _______________________ |

Derivative from First Principles | Page 45 | Exercise 4.1 | Q1,3,5,7 |

The Chain Rule | Page 48 | Exercise 4.3 | Q1a,d,2a,c,3b,4a,5a |

The Product Rule | Page 51 | Exercise 4.5 | Q1a-h,Q2b,Q3a-l |

The Quotient Rule | Page 52 | Exercise 4.6 | Q1,2,3,4 |

Differentiation - A Mixture! | Page 53 | Exercise 4.7 | Q1,2,3,4,5 |

Sec, Cosec & Cot | Page 55 | Exercise 4.8 | Q1a,b,2a,c,d,3a,c,e,g |

Exponential Functions | Page 58 | Exercise 4.9 | Q1a,c,e,2a,3e,4a,b,5a,e |

Logarithmic Functions | Page 58 | Exercise 4.9 | Q1k,m,o,q,s,2f,g,3a,b,c,4d,e,5d |

Nature & Sketching Polynomials | Page 70 | Exercise 5.5 | Q1a,b,c,2a,b |

Concavity | Page 73 | Exercise 5.7 | Q5a,b,c,Q1a,b |

Applications | Page 187 | Ex 11.1 | Q1a,b,e,f,2a,c,3a,c |

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**6. MIA Text Book Questions – Further Differentiation **

Recommended questions from the Maths In Action (2nd Edition) by Edward Mullan text book are shown below.

Worked solutions to all questions below are available in the Online Study Pack.

Subtopic _______________________________________ | Page Number _____________ | Exercise ______________ | Recommended Questions _____________________ |

Inverse Trig Functions & Chain Rule | Page 85 | Exercise 6.2 | Q1a,b,c,Q2b,c,dQ3a,d |

Inverse Trig Fns & Product/Quotient Rules | Page 86 | Exercise 6.3 | Q2,Q3 |

Implicit & Explicit Functions - 1 | Page 89 | Exercise 6.4 | Q1,Q2 |

Implicit & Explicit Functions - 2 | Page 89 | Exercise 6.4 | Q5,Q9,Q4 |

Second Derivatives of Implicit Functions | Page 90 | Exercise 6.5 | Q1a,d,f,k(i),6 |

Logarithmic Differentiation | Page 92 | Exercise 6.6 | Q1,Q2 |

Parametric Equations | Page 95 | Exercise 6.7 | Q1a,b,c |

Parametric Eqns - Differentiation | Page 96 | Exercise 6.8 | Q1,2,3 |

Parametric Eqns - Differentiation (Alternative) | Page 96 | Exercise 6.8 | Q1(i) |

Parametric Eqns - Differentiation (Alternative) | Page 96 | Exercise 6.8 | Q1(ii),Q2,Q3 |

Applications of Further Differentiation | Page 193 | Exercise 11.2 | Q1,Q2,Q3 |

**7. About the Nature of Turning Points**

For a more detailed explanation, please first read the Theory Guides above.

When sketching Polynomials we determine:

- Where the graph cuts the x and y-axis
- Where the stationary points occur (eg set f ‘ (x) = 0 and solve)
- How f(x) behaves as x approaches ± ∞
- The nature of the stationary points.

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At AH Maths to find the nature of the stationary points the second derivative is used.

Mr Smiley and Mr Frowny can help us remember the above:

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- Happy ⇒ Double Differentiate ⇒ Eyes Positive (f ”(x) > 0) and a Smile (Minimum TP)
- Frowny ⇒ Double Differentiate ⇒ Eyes Negative (f ”(x) < 0) and a Frown (Maximum TP)

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When the second derivative is zero there is no information about the nature of the stationary point and a nature table must be used to investigate,

Points of inflexion occur when the second derivative is zero.

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

Thanks to the SQA for making these available.

Worked solutions to SQA Exam Questions available in the Online Study Pack.

Year ____ | Paper Type _________________ | Exam Paper ______________ | Marking Scheme _______________________________________ |

2019 | AH Specimen | Specimen | Marking Scheme |

2019 | Advanced Higher | Exam Paper | Marking Scheme |

2018 | Advanced Higher | Exam Paper | Marking Scheme |

2017 | Advanced Higher | Exam Paper | Marking Scheme |

2016 | Advanced Higher | Exam Paper | Marking Scheme |

2016 | AH Specimen | Specimen | Marking Scheme |

2016 | AH Exemplar | Exemplar | Marking Scheme |

2015 | Advanced Higher | Exam Paper | Marking Scheme |

2014 | Advanced Higher | Exam Paper | Marking Scheme |

2013 | Advanced Higher | Exam Paper | Marking Scheme |

2012 | Advanced Higher | Exam Paper | Marking Scheme |

2011 | Advanced Higher | Exam Paper | Marking Scheme |

2010 | Advanced Higher | Exam Paper | Marking Scheme |

2009 | Advanced Higher | Exam Paper | Marking Scheme |

2008 | Advanced Higher | Exam Paper | Marking Scheme |

2007 | Advanced Higher | Exam Paper | Marking Scheme |

2006 | Advanced Higher | Exam Paper | Marking Scheme |

2005 | Advanced Higher | Exam Paper | Marking Scheme |

2004 | Advanced Higher | Exam Paper | Marking Scheme |

2003 | Advanced Higher | Exam Paper | Marking Scheme |

2002 | Advanced Higher | Exam Paper | Marking Scheme |

2001 | Advanced Higher | Exam Paper | Marking Scheme |

**9. AH Maths Past Paper Questions by Topic**

Thanks to the SQA for making these available.

Questions and answers have been split up by topic for ease of reference.

Worked solutions to all SQA questions below available in the Online Study Pack.

**10. AH Maths Text Book – Maths In Action (2nd Edition) by Edward Mullan **

A fully revised course for the new Curriculum for Excellence examination that is designed to fully support the course’s new structure and unit assessment. A part of the highly regarded Maths in Action series, it provides students with a familiar, clear and carefully structured learning experience that encourages them to build confidence and understanding.

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**11. Exam focused 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 the * Nature of Turning Points* to pass your final exam.

As well as students studying Advanced Higher Mathematics, the resources will benefit young adults studying A-Level Mathematics and undergraduates who need a little extra help. Particular benefit will be to students who have gained a ‘Conditional’ University place and are therefore required to pass in order to gain entry onto the course of their choice.

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 AH Maths course in 2020/21.

Topic Links

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