## Course description

## Further Mathematics Year 13 course 2: Applications of Differential Equations, Momentum, Work, Energy & Power, The Poisson Distribution, The Central Limit Theorem, Chi Squared Tests, Type I and II Errors

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 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.

## Upcoming start dates

1 start date available

## Training content

### 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.

- Momentum and the principle of conservation of momentum.
- 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

- Modelling elastic collision in two dimensions.
- Modelling inelastic collision in two dimensions.
- The kinetic energy lost in a collision.

- The Poisson distribution.
- Properties of the Poisson distribution.
- Approximating the binomial distribution.
- Testing for the mean of a Poisson distribution.

- The distribution of a sample mean.
- Underlying normal distributions.
- The Central Limit Theorem.

- Chi-squared tests and contingency tables.
- Fitting a theoretical distribution.
- Testing for goodness of fit.

- What are type I and type II errors?
- A summary of all probability distributions encountered in A level maths and further maths.

## Course delivery details

This course is offered through Imperial College London, a partner institute of EdX.

2-4 hours per week

## Costs

- Verified Track -$49
- Audit Track - Free

## Certification / Credits

### 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.
- How to solve coupled differential equations.
- How to calculate the impulse of one object on another in a collision.
- How to use the principle of conservation of momentum to model collisions in one dimension.
- How to use Newton’s experimental law to model inelastic collisions in one dimension.
- How to calculate the work done by a force and the work done against a resistive force.
- How to calculate gravitational potential energy and kinetic energy.
- How to calculate elastic potential energy.
- How to solve problems in which energy is conserved.
- How to solve problems in which some energy is lost through work against a dissipative force.
- How to calculate power and solve problems involving power.
- How to model elastic collision between bodies in two dimensions.
- How to model inelastic collision between two bodies in two dimensions.
- How to calculate the energy lost in a collision.
- How to calculate probability for a Poisson distribution.
- How to use the properties of a Poisson distribution.
- How to use a Poisson distribution to model a binomial distribution.
- How to use a hypothesis test to test for the mean of a Poisson distribution.
- How to estimate a population mean from sample data.
- How to estimating population variance using the sample variance. How to calculate and interpret the standard error of the mean.
- How and when to apply the Central Limit Theorem to the distribution of sample means.
- How to use the Central Limit Theorem in probability calculations, using a continuity correction where appropriate.
- How to apply the Central Limit Theorem to the sum of n identically distributed independent random variables.
- How to conduct a chi-squared test with the appropriate number of degrees of freedom to test for independence in a contingency table and interpret the results of such a test.
- How to fit a theoretical distribution, as prescribed by a given hypothesis involving a given ratio, proportion or discrete uniform distribution, to given data.
- How to use a chi-squared test with the appropriate number of degrees of freedom to carry out a goodness of fit test.
- How to calculate the probability of making a Type I error from tests based on a Poisson or Binomial distribution.
- How to calculate probability of making Type I error from tests based on a normal distribution.
- How to calculate P(Type II error) and power for a hypothesis test for tests based on a normal, Binomial or a Poisson distribution (or any other A level distribution).

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