#### COURSE DESCRIPTION

This course by Imperial College London is designed to help you develop the skills you need to succeed in your A-level further maths exams.

You will investigate key topic areas to gain a deeper understanding of the skills and techniques that you can apply throughout your A-level study. These skills include:

- Fluency – selecting and applying correct methods to answer with speed and efficiency
- Confidence – critically assessing mathematical methods and investigating ways to apply them
- Problem solving – analysing the ‘unfamiliar’ and identifying which skills and techniques you require to answer questions
- Constructing mathematical argument – using mathematical tools such as diagrams, graphs, logical deduction, mathematical symbols, mathematical language, construct mathematical argument and present precisely to others
- Deep reasoning – analysing and critiquing mathematical techniques, arguments, formulae and proofs to comprehend how they can be applied

Over eight modules, you will be introduced:

- The determinant and inverse of a 3 x 3 matrix
- Mathematical induction
- Differentiation and integration methods and some of their applications
- Maclaurin series
- DeMoivre’s Theorem for complex numbers and their applications
- Polar coordinates and sketching polar curves
- Hyperbolic functions

Your initial skillset will be extended to give a clear understanding of how background knowledge underpins the A-level further mathematics course. You’ll also, be encouraged to consider how what you know fits into the wider mathematical world.

#### LEARNING OUTCOMES

- How to find the determinant of a complex number without using a calculator and interpret the result geometrically.
- How to use properties of matrix determinants to simplify finding a determinant and to factorise determinants.
- How to use a 3 x 3 matrix to apply a transformation in three dimensions
- How to find the inverse of a 3 x 3 matrix without using a calculator.
- How to prove series results using mathematical induction.
- How to prove divisibility by mathematical induction.
- How to prove matrix results by using mathematical induction.
- How to use the chain, product and quotient rules for differentiation.
- How to differentiate and integrate reciprocal and inverse trigonometric functions.
- How to integrate by inspection.
- How to use trigonometric identities to integrate.
- How to use integration methods to find volumes of revolution.
- How to use integration methods to find the mean of a function.
- How to express functions as polynomial series.
- How to find a Maclaurin series.
- How to use standard Maclaurin series to define related series.
- How to use De Moivre’s Theorem.
- How to use polar coordinates to define a position in two dimensional space.
- How to sketch the graphs of functions using polar coordinates.
- How to define the hyperbolic sine and cosine of a value.
- How to sketch graphs of hyperbolic functions.
- How to differentiate and integrate hyperbolic functions.

#### Syllabus

**Module 1: Matrices – The determinant and inverse of a 3 x 3 matrix**

- Moving in to three dimensions
- Conventions for matrices in 3D
- The determinant of a 3 x 3 matrix and its geometrical interpretation
- Determinant properties
- Factorising a determinant
- Transformations using 3 x 3 matrices
- The inverse of a 3 x 3 matrix

**Module 2: Mathematical induction**

- The principle behind mathematical induction and the structure of proof by induction
- Mathematical induction and series
- Proving divisibility by induction
- Proving matrix results by induction

**Module 3: Further differentiation and integration**

- The chain rule
- The product rule and the quotient rule
- Differentiation of reciprocal and inverse trigonometric functions
- Integrating trigonometric functions
- Integrating functions that lead to inverse trigonometric integrals
- Integration by inspection
- Integration using trigonometric identities

**Module 4: Applications of Integration**

- Volumes of revolution
- The mean of a function

**Module 5: An Introduction to Maclaurin series**

- Expressing functions as polynomial series from first principles
- Maclaurin series
- Adapting standard Maclaurin series

**Module 6: Complex Numbers: De Moivre’s Theorem and exponential form**

- De Moivre’s theorem and it’s proof
- Using de Moivre’s Theorem to establish trigonometrical results
- De Moivre’s Theorem and complex exponents

**Module 7: An introduction to polar coordinates**

- Defining position using polar coordinates
- Sketching polar curves
- Cartesian to polar form and polar to Cartesian form

**Module 8: Hyperbolic functions**

- Defining hyperbolic functions
- Graphs of hyperbolic functions
- Calculations with hyperbolic functions
- Inverse hyperbolic functions

* Differentiating and integrating hyperbolic functions

### Course Features

- Lectures 0
- Quizzes 0
- Duration 7 weeks
- Skill level All levels
- Language English
- Students 0
- Assessments Yes