Professional Course

# Further Mathematics Year 13 course 2

edX, Online
Length
7 weeks
Price
49 USD
Next course start
Start anytime See details
Delivery
Self-paced Online
Length
7 weeks
Price
49 USD
Next course start
Start anytime See details
Delivery
Self-paced Online
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## 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

#### Start anytime

• Self-paced Online
• Online
• English

## 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.
Module 2: Momentum and Impulse
• 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
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.

## 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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