#### 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, the 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 to

- Analytical and numerical methods for solving first-order differential equations
- The nth roots of unity, the nth roots of any complex number, geometrical applications of complex numbers.
- Coordinate systems and curve sketching.
- Improper integrals, integration using partial fractions and reduction formulae
- The area enclosed by a curve defined by parametric equations or polar equations, arc length and the surface area of revolution.
- Solving second-order differential equations
- The vector product and its applications
- Eigenvalues, eigenvectors, diagonalization and the Cayley-Hamilton Theorem.

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 general or particular solution to a first-order differential equation by inspection or by using an integrating factor.
- How to find a numerical solution to a differential equation using the Euler method or an improved Euler method..
- How to find the nth roots of unity
- How to find the nth roots of a complex number in the form
- How to use complex roots of unity to solve geometrical problems.
- How to identify the features of parabolas, rectangular hyperbolae, ellipses and hyperbolae defined by Cartesian and parametric equations.
- How to identify features of graphs defined by rational functions.
- How to define a parabola, ellipse or hyperbola using focus-directrix properties and eccentricity.
- How to evaluate improper integrals.
- How to integrate using partial fractions
- How to derive and use reduction formulae
- How to find areas enclosed by curves that are defined parametrically.
- How to find the area enclosed by a polar curve.
- How to calculate arc length.
- How to calculate the surface area of revolution.
- How to find the auxiliary equation for a second order differential equation.

#### Syllabus

**Module 1: First Order Differential Equations**

- Solving first order differential equations by inspection
- Solving first order differential equations using an integrating factor
- Finding general and particular solutions of first-order differential equations
- Euler’s method for finding the numerical solution of a differential equation
- Improved Euler methods for solving differential equations.

**Module 2: Further Complex Numbers**

- The nth roots of unity and their geometrical representation
- The nth roots of a complex number and their geometrical representation
- Solving geometrical problems using complex numbers.

**Module 3: Properties of Curves**

- Cartesian and parametric equations for the parabola and rectangular hyperbola, ellipse and hyperbola.
- Graphs of rational functions
- Graphs of , , for given
- The focus-directrix properties of the parabola, ellipse and hyperbola, including the eccentricity.

**Module 4: Further Integration Methods**

- Evaluate improper integrals where either the integrand is undefined at a value in the range of integration or the range of integration extends to infinity.
- Integrate using partial fractions including those with quadratic factors in the denominator
- Selecting the correct substitution to integrate by substitution.
- Deriving and using reduction formula

**Module 5: Further Applications of Integration**

- Finding areas enclosed by curves that are defined parametrically
- Finding the area enclosed by a polar curve
- Using integration methods to calculate the arc length
- Using integration methods to calculate the surface area of revolution

**Module 6: Second Order Differential Equations**

- Solving differential equations of form y″ + ay′ + by = 0 where a and b are constants by using the auxiliary equation.
- Solving differential equations of form y ″+ a y ′+ b y = f(x) where a and b are constants by solving the homogeneous case and adding a particular integral to the complementary function

**Module 7: The Vector (cross) Product**

- The definition and properties of the vector product
- Using the vector product to calculate areas of triangles.
- The vector triple product.
- Using the vector triple product to calculate the volume of a tetrahedron and the volume of a parallelepiped
- The vector product form of the vector equation of a straight line
- Solving geometrical problems using the vector product

**Module 8: Matrices – Eigenvalues and Eigenvectors**

- Calculating eigenvalues and eigenvectors of 2 × 2 and 3 × 3 matrices.
- Reducing matrices to diagonal form.
- Using the Cayley-Hamilton Theorem

### Course Features

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