Develop your thinking skills, fluency and confidence in the applied mathematics content of A-level further maths and prepare for undergraduate STEM degrees. 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.
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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 to
- Simple harmonic motion and damped oscillations.
- Impulse and momentum.
- The work done by a constant and a variable force, kinetic and potential energy (both gravitational and elastic) conservation of energy, the work-energy principle, conservative and dissipative forces, power.
- Oblique impact for elastic and inelastic collision in two dimensions.
- The Poisson distribution, its properties, approximation to a binomial distribution and hypothesis testing.
- The distribution of sample means and the central limit theorem.
- Chi-squared tests, contingency tables, fitting a theoretical distribution and goodness of fit.
- Type I and type II errors in statistical tests.
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.
What you'll learn
- How to derive and solve a second order differential equation that models simple harmonic motion.
- How to derive a second order differential equation for damped oscillations.
- The meaning of underdamping, critical damping and overdamping.
Syllabus
Module 1: Applications of Differential Equations
Using differential equations in modelling in kinematics and in other contexts.
Hooke’s law.
Simple harmonic motion (SHM).
Damped oscillatory motion.
Light, critical and heavy damping.
Coupled differential equations.
Module 2: Momentum and Impulse
Momentum and the principle of conservation of momentum.
Impulse.
Newton’s experimental law (restitution)
Impulse for variable forces.
Module 3: Work, Energy and Power
The work-energy principle.
Conservation of mechanical energy.
Gravitational potential energy and kinetic energy.
Elastic potential energy.
Conservative and dissipative forces.
Power
Module 4: Oblique Impact
Modelling elastic collision in two dimensions.
Modelling inelastic collision in two dimensions.
The kinetic energy lost in a collision.
Module 5: Expectation and Variance and the Poisson Distribution
The Poisson distribution.
Properties of the Poisson distribution.
Approximating the binomial distribution.
Testing for the mean of a Poisson distribution.
Module 6: The Central Limit Theorem
The distribution of a sample mean.
Underlying normal distributions.
The Central Limit Theorem.
Module 7: Chi-Squared Tests
Chi-squared tests and contingency tables.
Fitting a theoretical distribution.
Testing for goodness of fit.
Module 8: Type I and Type II Errors
What are type I and type II errors?
A summary of all probability distributions encountered in A level maths and further maths.
